\jP Biodiversity
fe^HeriUge
http://www.biodiversitylibrary.org
The Bell System technical journal.
[Short Hills, N.J., etc.,American Telephone and Telegraph Co.]
http://www.bi0diversitylibrary.0rg/bibli0graphy/l 81 81
27: http://www.biodiversitylibrary.org/item/57782
Page(s): Page 379, Page 380, Page 381 , Page 382, Page 383, Page 384, Page 385, Page 386,
Page 387, Page 388, Page 389, Page 390, Page 391 , Page 392, Page 393, Page 394, Page
395, Page 396, Page 397, Page 398, Page 399, Page 400, Page 401 , Page 402, Page 403,
Page 404, Page 405, Page 406, Page 407, Page 408, Page 409, Page 41 0, Page 41 1 , Page
41 2, Page 41 3, Page 41 4, Page 41 5, Page 41 6, Page 41 7, Page 41 8, Page 41 9, Page 420,
Page 421 , Page 422, Page 423, Page 623, Page 624, Page 625, Page 626, Page 627, Page
628, Page 629, Page 630, Page 631 , Page 632, Page 633, Page 634, Page 635, Page 636,
Page 637, Page 638, Page 639, Page 640, Page 641 , Page 642, Page 643, Page 644, Page
645, Page 646, Page 647, Page 648, Page 649, Page 650, Page 651 , Page 652, Page 653,
Page 654, Page 655, Page 656
Contributed by: Prelinger Library (archive.org)
Sponsored by: Internet Archive
Generated 26 April 201 1 0:35 AM
http://www.biodiversitylibrary.org/pdf2/002916500057782
This page intentionally left blank.
i
The Bell System Technical Journal
Vol. XXVII July, 194S No, 3
A Mathematical Theory of Communication
By C. E. SHANNON
iNTRODtJCTION
I
THE recent development of various methods of modulation such as PCM
and PPM which exchange bandwidth for signal-to-noise ratio has in-
tensified the interest in a general theory of communication. A basis for
such a theory is contained in the important papers of Nyquist^ and Hartley^
on this subject. In the present paper we will extend the theor^^ to include a
number of new factors^ in particular the effect of noise in the channel, and
the savings possible due to the statistical structure of the original message
and due to the nature of the final destination of the information.
The fundamental problem of communication is that of reproducing at
one point either exactly or approximately a message selected at another
point. Frequently the messages have meaning; that is they refer to or are
correlated according to some system with certain physical or conceptual
entities. These semantic aspects of communication are irrelevant to the
engineering problem. The significant aspect is that the actual message is
one selected from a set of possible messages. The system must be designed
to operate for each possible selection, not just the one which will actually
be chosen since this is unknown at the time of design.
If the number of messages in the set is finite then this number or any
monotonic function of this number can be regarded as a measure of the in-
formation produced when one message is chosen from the set, all choices
being equally likely. As was pointed out by Hartley the most natural
choice is the logarithmic function. Although this definition must be gen-
eralized considerably when we consider the influence of the statistics of the
message and when we have a continuous range of messages, we will in all
cases use an essentially logarithmic measure.
The logarithmic measure is more convenient for various reasons:
1. It is practically more useful. Parameters of engineering importance
^ Nyquist, H., "Certain Factors Affecting Telegraph Speed,*' BellSystem Technical Jour-
nal, April 1924, p. 324; "Certain Topics in Telegraph Transmission Theory," ^4. /. E. E.
Trans., v. 47, April 1928, p. 617.
* Hartley, R. V. L., "Transmission of Information," Bdl System Technical Journal, July
1928, p. 535.
379
380 BELL SYSTEM TECHNICAL JOURNAL
such as time, bandwidth, number of relays, etc., tend to vary linearly with
the logaritlini of the number of possibilities. For example, adding one relay
to a group doubles the number of possible states of the relays. It adds 1
to the base 2 logarithm of this number. Doubling the time roughly squares
the number of possible messages, or doubles the logarithm, etc.
2. It is nearer to our intuitive feeling as to the proper measure. This is
closely related to (1) since we intuitively measure entities by linear com-
parison with common standards. One feels, for example, that two punched
cards should have twice the capacity of one for information storage, and two
identical channels twice the capacity of one for transmitting information.
3. It is mathematically more suitable. Many of the limiting operations
are simple in terms of the logarithm but would require clumsy restatement in
terms of the number of possibilities.
The choice of a logarithmic base corresponds to the choice of a unit for
measuring information. If the base 2 is used the resulting units may be
called binary digits, or more briefly bits^ a word suggested by J. W. Tukey.
A device with two stable positions, such as a relay or a flip-flop circuit, can
store one bit of information. N such devices can store N bits, since the
total number of possible states is 2^ and log22^ = N, If the base 10 is
used the units may be called decimal digits. Since
log2 M = logio Jli'/logio2
4
= 3.32 logio Mf
a decimal digit is about 3| bits. A digit wheel on a desk computing machine
has ten stable positions and therefore has a storage capacity of one decimal
*
digit. In analytical work where integration and differentiation are involved
the base e is sometimes useful. The resulting units of information will be
called natural units. Change from the base a to base b merely requires
multiplication by log6 a.
By a communication system we will mean a system of the type indicated
schematically in Fig. 1. It consists of essentially five parts:
1. An informalwn source which produces a message or sequence of mes-
sages to be communicated to the receiving terminal. The message may be
of various tyi)es: e.g. (a) A sequence of letters as in a telegraph or teletype
system ; (b) A single function of time fit) as in radio or telephony ; (c) A
function of time and other variables as in black and white television — ^here
the message may be thought of as a function /(:*:, y, /) of two space coordi-
nates and time, the light intensity at point f^, y) and time / on a pickup tube
plate; (d) Two or more functions of time, say f(l), g(/), //(/) — this is the
case in "three dimensional" sound transmission or if the system is intended
to ser\nce several individual channels in multiplex; (e) Several functions of
MATHEMATICAL THEORY OF COMMUNICATION
381
several variables — in color television the message consists of three functions
/(^j y> Oj s(^i yj 0> K^j y^ defined in a three-dimensional continuum —
we may also think of these three functions as components of a vector field
defined in the region — similarly, several black and white television sources
would produce "messages" consisting of a number of functions of three
variables; (f) Various combinations also occur, for example in television
with an associated audio channel.
2. A trausmhter which operates on the message in some way to produce a
signal suitable for transmission over the channel. In telephony this opera-
tion consists merely of changing sound pressure into a proportional electrical
current. In telegraphy we have an encoding operation which produces a
sequence of dots, dashes and spaces on the channel corresponding to the
message. In a multiplex PCM system the different speech functions must
be sampled, compressed, quantized and encoded, and finally interleaved
INFORMATrON
SOURCE TRANSMITTER
RECEIVER
OeST I NATION
SIGNAL
MESSAGE
RECEIVED
SIGNAL
MESSAGE
NO ISE
SOURCE
Fig. 1 — Schematic diagram of a general communication system.
properly fo construct the signal. Vocoder systems, television, and fre-
quency modulation are other examples of complex operations applied to the
message to obtain the signal.
3. The channel is merely the medium used to transmit the signal from
transmitter to receiver. It may be a pair of wires, a coaxial cable, a band of
radio frequencies, a beam of light, etc.
4. The receiver ordinarily performs the inverse operation of that done by
the transmitter, reconstructing the message from the signal.
5. The desii nation is the person (or thing) for whom the message is in-
tended.
We wish to consider certain general problems involving communication
systems. To do this it is first necessary to represent the various elements
involved as mathematical entities, suitably idealized from their physical
counterparts. We may roughly classify communication systems into three
main categories: discrete, continuous and mixed. By a discrete system we
will mean one in which both the message and the signal are a sequence of
382 BELL SYSTEM TECHNICAL JOURNAL
discrete symbols. A typical case is telegraphy where the message is a
sequence of letters and the signal a sequence of dots, dashes and spaces.
A continuous system is one in which the message and signal are both treated
as continuous functions, e.g. radio or television. A mixed system is one in
which both discrete and continuous variables appear, e.g., PCM transmis-
sion of S]
We first consider the discrete case. This case has applications not only
in communication theory, but also in the theory of computing machines,*
the design of telephone exchanges and other fields. In addition the discrete
case forms a foundation for the continuous and mixed cases which will be
treated in the second half of the paper.
PART I: DISCRETE NOISELESS SYSTEMS
1. The Discrete Noiseless Channel
Teletype and telegraphy are two simple examples of a discrete channel
for transmitting information. Generally, a discrete channel will mean a
system whereby a sequence of choices from a finite set of elementary sym-
bols 5i • • * 5ft can be transmitted from one point to another. Each of the
symbols Si is assumed to have a certain duration in time ti seconds (not
necessarily the same for different Si , for example the dots and dashes in
telegraphy). It is not 'required that all possible sequences of the Si be cap-
able of transmission on the system ; certain sequences only may be allowed.
These will be possible signals for the channel. Thus in telegraphy suppose
the symbols are: (1) A dot, consisting of line closure for a unit of time and
then line open for a unit of time; (2) A dash, consisting of three time units
of closure and one unit open ; (3) A letter space consisting of, say, three units
of line open; (4) A word space of six units of line open. We might place
the restriction on allowable sequences that no spaces follow each other (for
if two letter spaces are adjacent, it is identical with a word space). The
question we now consider is how one can measure the capacity of such a
channel to transmit information.
In the teletype case where all symbols are of the same duration, and any
sequence of the 32 symbols is allowed the answer is easy. Each symbol
represents five bits of information. If the system transmits n symbols
per second it is natural to say that the channel has a capacity of Sn bits per
second. This does not mean that the teletype channel will always be trans-
mitting information at this rate — this is the maximum possible rate and
whether or not the actual rate reaches this maximum depends on the source
of information which feeds the channel, as will appear later.
MATHEMATICAL THEORY OF COMMUNICATION 383
In the more general case with different lengths of symbols and constraints
on the allowed sequences, we make the following definition:
Definition: The capacity C of a discrete channel is given by
^ ,. logiV(r)
C = Lim — ^
00
T
where N(T) is the number of allowed signals of duration T.
It is easily seen that in the teletype case this reduces to the previous
result. It can be shown that the limit in question will exist as a finite num-
ber in most cases of interest. Suppose all sequences of the symbols 5i , ♦ • • ,
Sn are allowed and these symbols have durations /i ,-••,/« . What is the
channel capacity? If N(t) represents the number of sequences of duration
t we have
N{1) = N{1 - h) -^ N{t- h) + '-• + N{t - tn)
The total number is equal to the sum of the numbers of sequences ending in
SijS2i ' • ' J Sn and these are N(t — ii)j N{t — /2), • • • , N{t — /„), respec-
tively. According to a well known result in finite differences, N{t) is then
asymptotic for large / to Xq where Xq is the largest real solution of the
characteristic equation :
■*! I V — *2 I I "V — '
XT'' + X~'' + h A " - 1
and therefore
C = log Xo
In case there are restrictions on allowed sequences we may still often ob-
tain a difference equation of this type and find C from the characteristic
equation. In the telegraphy case mentioned above
Nil) = N{t - 2) + Nit - 4) + N{t - 5) + N{t - 7) + NQ - 8)
+ N{i - 10)
as we see by counting sequences of symbols according to the last or next to
the last symbol occurring. Hence C is — log mo where /uo is the positive
root of 1 = M^ + ju* + M^ + /*' + M^ + M^^' Solving this we find C = 0.539.
A very general t>'pe of restriction which may be placed on allowed se-
quences is the following : We imagine a number of possible states ai , a2 , • • • ,
Om . For each state only certain symbols from the set 5i , • • • , 5» can be
transmitted (different subsets for the different states). When one of these
has been transmitted the state changes to a new state depending both on
the old state and the particular symbol transmitted. The telegraph case is
a simple example of this. There are two states depending on whether or not
384 BELL SYSTEM TECHNICAL JOURNAL
a space Vas the last symbol transmitted. If so then only a dot or a dash
can be sent next and the state always changes. If not, any symbol can be
transmitted and the state changes if a space is sent, otherwise it remains
the same. The conditions can be indicated in a linear graph as shown in
Fig. 2. The junction points correspond to the states and the lines indicate
the symbols possible in a state and the resulting state. In Appendix I it is
shown that if the conditions on allowed sequences can be described in this
form C will exist and can be calculated in accordance with the following
result:
Theorem 1: Let hi) be the duration of the s^^ symbol which is allowable in
state i and leads to state j. Then the channel capacity C is equal to log
W where W is the largest real root of the determinant equation:
Y.w-^'i^^ - hii\ = 0.
j;
where 8ij = 1 if i = j and is zero otherwise
DASH
DOT
DASH
WORD SPACE
Fig. 2 — Graphical representation of the constraints on telegraph symbols
For example, in the telegraph case (Fig. 2) the determinant is:
- 1 (W'"" + W~^)
{W~^ + W') (W"^ + IT^* - 1)
=
On expansion this leads to the equation given above for this case.
2. The Discrete Source of Information
We have seen that under very general conditions the logarithm of the
number of possible signals in a discrete channel increases linearly with time.
The capacity to transmit information can be specified by giving this rate of
increase, the number of bits per second required to specify the particular
signal used.
We now consider the information source. How is an information source
to be described mathematically, and how much information in bits per sec-
ond is produced in a given source? The main point at issue is the effect of
statistical knowledge about the source in reducing the required capacity
MATHEMATICAL THEORY OF COMMUNICATION 385
of the channel, by the use of proper encoding of the information. In teleg-
raphy, for example, the messages to be transmitted consist of sequences
of letters. These sequences, however, are not completely random. In
general, they form sentences and have the statistical structure of, say, Eng-
lish. The letter E occurs more frequently than Q, the sequence TH more
frequently than XP, etc. The existence of this structure allows one to
make a saving in time (or channel capacity) by properly encoding the mes-
sage sequences into signal sequences. This is already done to a limited ex-
tent in telegraphy by using the shortest channel symbol, a dot^ for the most
common English letter E; while the infrequent letters, Q, X, Z are repre-
sented by longer sequences of dots and dashes. This idea is carried still
further in certain commercial codes where common words and phrases are
represented by four- or five-letter code groups with a considerable saving in
average time. The standardized greeting and amiiversar}^ telegrams now
in use extend this to the point of encoding a sentence or two into a relatively
short sequence of numbers.
We can think of a discrete source as generating the message, symbol by
S)rmbol. It will choose successive symbols according to certain probabilities
depending, in general, on preceding choices as well as the particular symbols
in question. A physical system, or a mathematical model of a system which
produces such a sequence of symbols governed by a set of probabilities is
known as a stochastic process.^ \\'e may consider a discrete source, there-
fore, to be represented by a stochastic process. Conversely, any stochastic
process which produces a discrete sequence of symbols chosen from a finite
set may be considered a discrete source. This will include such cases as:
1. Natural written languages such as English, German, Chinese.
2. Continuous information sources that have been rendered discrete by some
quantizing process. For example, the quantized speech from a PCM
transmitter, or a quantized television signal.
3. Mathematical cases where we merely define abstractly a stochastic
process w^hich generate a sequence of symbols. The following are ex-
amples of this last type of source.
(A) Suppose we have five letters A, B, C, D, E which are chosen each
with probabihty .2, successive choices being independent. This
would lead to a sequence of which the following is a typical example.
BDCBCECCCADCBDDAAECEEA
ABBDAEECACEEBAEECBCEAD
This was constructed with the use of a table of random numbers."*
' See, for example, S. Chandrasekhar, "Stochastic Problems in Physics and Astronomy,"
Reviews of Modern Physics, v. 15, No. 1, January 1943, p. 1.
^ Kendall and Smith, "Tables of Random Sampling Xumbers," Cambridge, 1939.
386
BELL SYSTEM TECHNICAL JOURNAL
(B) Using the same five letters let the probabiUties be .4, .1, .2, .2, .1
respectively, with successive choices independent. A typical
message from this source is then:
AAACD CBDCEAAD ADACED A
E AD CAB ED ADD CEC AAA A AD " r
(C) A more complicated structure is obtained if successive symbols are
not chosen independently but their probabilities depend on preced-
ing letters. In the simplest case of this type a choice depends only
on the preceding letter and not on ones before that. The statistical
structure can then be described by a set of transition probabilities
piij) J the probability that letter i is followed by letter j. The in-
dices i and j range over all the possible symbols. A second equiv-
alent way of specifying the structure is to give the ''digram" prob-
abilities p(iyj)f i.e., the relative frequency of the digram ij. The
letter frequencies />(i), (the probability of letter i), the transition
probabilities pi(j) and the digram probabilities p{ij j) are related by
the following formulas.
P(i) = L pa, j) = Z PU, i) = L p{j)pAi)
piuj) = pO)pi(J)
Z pi(j) = E pi^ = Z pa, i) = 1 .
».;
As a specific example suppose there are three letters A, B, C with the prob
ability tables:
pif)
piij)
•
J
A
B
c
A
i
1
i B
i
1
5
C
1
2
1
1
10
A
B
C
9
16
ST
2
pih j)
•
J
1
A
B
c
A
4
13^
1
1&
i B
8
27
^
C
1
ST
A
1
133
A typical message from this source is the following:
ABBABABABABABABBBABBBBBAB
ABABABABBBACACABBABBBBABB
ABACBBBABA
The next increase in complexity would involve trigrani frequencies
but no more. The choice of a letter would depend on the preceding
two letters but not on the message before that point. A set of tri-
gram frequencies p(i, y, k) or equivalently a set of transition prob-
MATHEMATICAL THEORY OF COMMUNICATION 387
abilities pij{k) would be required. Continuing in this way one ob-
tains successively more complicated stochastic processes. In the
general w-gram case a set of «-gram probabilities p(ii , t2 » * • • j in)
or of transition probabilities ^», , ta , ■. , t„-iOn) is required to
specify the statistical structure,
(D) Stochastic processes can also be defined which produce a text con-
sisting of a sequence of "words." Suppose there are five letters
A, B, C, D, E and 16 "words" in the language with associated
probabilities :
.10 A .16 BEBE .11 CABED .04 DEB
.04 ADEB .04 BED .05 CEED .15 DEED
,05ADEE .02BEED .08 DAB ,01 EAB
.OIBADD ,05 CA .04 DAD .05 EE
Suppose successive "words" are chosen independently and are
separated by a space. A typical message might be:
DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBE
BEBE BEBE ADEE BED DEED DEED CEED x\DEE A DEED
DEED BEBE CABED BEBE BED DAB DEED ADEB
If all the words are of finite length this process is equivalent to one
of the preceding type, but the description may be simpler in terms
of the word structure and probabilities. We may also generalize
here and introduce transition probabilities between words, etc.
These artificial languages are useful in constructing simple problems and
examples to illustrate various possibilities. We can also approximate to a
natural language by means of a series of simple artificial languages. The
zero-order approximation is obtained by choosing all letters with the same
probability and independently. The first-order approximation is obtained
by choosing successive letters independently but each letter having the
same probability that it does in the natural language.* Thus, in the first-
order approximation to English, E is chosen with probability .12 (its fre-
quency in normal English) and W with probability .02, but there is no in-
fluence between adjacent letters and no tendency to form the preferred
digrams such as TH, ED, etc. In the second-order approximation, digram
structure is introduced. After a letter is chosen, the next one is chosen in
accordance with the frequencies with which the various letters follow the
first one. This requires a table of digram frequencies p»(;). In the third-
order approximation, trigram structure is introduced. Each letter is chosen
with probabilities which depend on the preceding two letters.
6 Letter, digram and trigram frequencies are given in "Secret and Urgent" by Fletcher
Pratt, Blue Ribbon Books 1939. Word frequencies are tabulated in "Relative Frequency
of English Speech Sounds," G, Dewey, Harvard University Press, 1923.
m BELL SYSTEM TECHNICAL JOURNAL
3. The Series of Approximations to English
To give a visual idea of how this series of processes approaches a lailgiiagej
typical sequences in the approximations to English have been constructed
and are given below. In all cases we have assumed a 27-symbol "alphabet,"
the 26 letters and a space.
1. Zero-order approximation (symbols independent and equi-probable).
XFOML RXKHRJFFJUJ ZLPWCFWKCYJ
FFJEYVKCQSGXYD QPAAMKBZAACIBZLHJQD
2. First-order approximation (symbols independent but with frequencies
of English text).
OCRO HLI RGWR NMIELWIS EU LL NBNESEBYA TH EEI
ALHENHTTPA OOBTTVA NAH BRL
3. Second-order approximation (digram structure as in English).
ON IE ANTSOUTINYS ARE T INCTORE ST BE S DEAMY
ACHIN D ILONASIVE TUCOOWE AT TEASONARE FUSO
TIZIN ANDY TOBE SEACE CTISBE
4. Third-order approximation (trigram structure as in English).
IN NO 1ST LAT WHEY CRATICT FROURE BTRS GROCID
PONDEXOME OF DEMONSTURES OF THE REPTAGIN IS
REGOACTIONA OF CRE
5. First-Order Word Approximation. Rather than continue with tetra-
gram, • • • , w-gram structure it is easier and better to jump at this
point to word units. Here words are chosen independently but with
their appropriate frequencies.
REPRESENTING AND SPEEDILY IS AN GOOD APT OR
COME CAN DIFFERENT NATURAL HERE HE THE A IN
CAME THE TO OF TO EXPERT GRAY COME TO FUR-
NISHES THE LINE MESSAGE HAD BE THESE.
6. Second-Order Word Approximation. The word transition probabil-
ities are correct but no further structure is included.
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH
WRITER THAT THE CHARACTER OF THIS POINT IS
THEREFORE ANOTHER METHOD FOR THE LETPERS
THAT THE TIME OF WHO E\ ER TOLD THE PROBLEM
FOR AN UNEXPECTED
The resemblance to ordinary English text increases quite noticeably at
each of the above steps. Note that these samples have reasonably good
structure out to about twice the range that is taken into account in their
construction. Thus in (3) the statistical process insures reasonable text
for two-letter sequence, but four-letter sequences from the sample can
usually be fitted into good sentences. In (6) sequences of four or more
MATHEMATICAL THEORY OF COMMUNICATION 389
words can easily be placed in sentences without unusual or strained con-
structions. The particular sequence of ten words * 'attack on an English
writer that the character of this" is not at all unreasonable. It appears
then that a sufficiently complex stochastic process will give a satisfactory
representation of a discrete source.
The first two samples w^ere constructed by the use of a book of random
numbers in conjunction with (for example 2) a table of letter frequencies.
This method might have been continued for (3), (4), and (5), since digram,
trigram, and word frequency tables are available, but a simpler equivalent
method was used. To construct (3) for example, one opens a book at ran-
dom and selects a letter at random on the page. This letter is recorded.
The book is then opened to another page and one reads until this letter is
encountered. The succeeding letter is then recorded. Turning to another
page this second letter is searched for and the succeeding letter recorded,
etc. A similar process was used for (4), (5), and (6). It would be interest-
ing if further approximations could be constructed, but the labor involved
becomes enormous at the next stage.
4. Graphical Representation of a Markopf Process
Stochastic processes of the type described above are known mathe-
matically as discrete Markoflf processes and have been extensively studied in
the literature.^ The general case can be described as follows: There exist a
finite number of possible "states" of a system; 5i , 52 , • • • , 5„ , In addi-
tion there is a set of transition probabilities; pi(j) the probability that if the
system is in state Si it will next go to state Sj . To make this Markoff
process into an information source we need only assume that a letter is pro-
duced for each transition from one state to another. The states will corre-
spond to the ''residue of influence" from preceding letters.
The situation can be represented graphically as shown in Figs. 3, 4 and 5.
The "states" are the junction points in the graph and the probabilities and
letters produced for a transition are given beside the corresponding line.
Figure 3 is for the example B in Section 2, while Fig. 4 corresponds to the
example C. In Fig. 3 there is only one state since successive letters are
independent. In Fig. 4 there are as many states as letters. If a trigram
example were constructed there would be at most n^ states corresponding
to the possible pairs of letters preceding the one being chosen. Figure 5
is a graph for the case of word structure in example D. Here S corresponds
to the "space" symbol.
* For a detailed treatment see M. Frechet, "Methods des fonctions arbitraires. Theorie
des enenements en chaine dans !e cas d'un nombre fini d'etats possibles." Paris, Gauthier-
Villars, 1938.
590
BELL SYSTEM TECHNICAL JOURNAL
5. Ergodic and Mixed Sources
As we have indicated above a discrete source for our purposes can be con-
sidered to be represented by a Markon process. Among the possible discrete
Markoff processes there is a group with special properties of significance in
D .2
Fig. 3 — ^A graph corresponding to the source in example B.
Fig. 4 — A graph corresponding to the source in example C,
Fig. 5 — A graph corresponding to the source in example D.
*
communication theory. This special class consists of the ''ergodic'* proc-
esses and we shall call the corresponding sources ergodic sources. Although
a rigorous definition of an ergodic process is somewhat involved, the general
idea is simple. In an ergodic process every sequence produced by the proc-
MATHEMATICAL THEORY OF COMMUNICATION 391
ess is the same in statistical properties. ' Thus the letter frequencies,
digram frequencies, etc., obtained from particular sequences will, as the
lengths of the sequences increase, approach definite limits independent of
the particular sequence. Actually this is not true of every sequence but the
set for which it is false has probability zero. Roughly the ergodic property
means statistical homogeneity.
All the examples of artificial languages given above are ergodic. This
property is related to the structure of the corresponding graph. If the graph
has the following two properties^ the corresponding process will be ergodic:
1 . The graph does not consist of two isolated parts A and B such that it is
impossible to go from junction points in part A to junction points in
part B along lines of the graph in the direction of arrows and also im-
possible to go from junctions in part B to junctions in part A.
2. A closed series of hues in the graph with all arrows on the lines pointing
in the same orientation will be called a "circuit." The "length" of a
circuit is the number of lines in it. Thus in Fig. 5 the series BEBES
is a circuit of length 5. The second property required is that the
greatest common divisor of the lengths of all circuits in the graph be
one.
If the first condition is satisfied but the second one violated by having the
greatest common divisor equal to d > 1, the sequences have a certain type
of periodic structure. The various sequences fall into d different classes
which are statistically the same apart from a shift of the origin (i.e., which
letter in the sequence is called letter 1). By a shift of from up to d — 1
any sequence can be made statistically equivalent to any other. A simple
example with d = 2 is the following: There are three possible letters a, 6, c.
Letter a is followed with either 6 or c with probabilities | and f respec-
tively. Either & or c is always followed by letter a. Thus a typical sequence
IS
ab ac a c a c ab ac ab a b ac a c
This type of situation is not of much importance for our work.
If the first condition is violated the graph may be separated into a set of
subgraphs each of which satisfies the first condition. We will assume that
the second condition is also satisfied for each subgraph. We have in this
case what may be called a "mixed" source made up of a number of pure
components. The components correspond to the various subgraphs.
If Li , La , L3 , • • • are the component sources we may write
L = piLi + P2L2 + paLz +
A *
where pi is the probability of the component source Li .
' These are restatements in terms of the graph of conditions given in Frechet.
392 BELL SYSTEM TECHNICAL JOURNAL
Physically the situation represented is this: There are several different
sources Li ^ L^ , L^ , • • • which are each of homogeneous statistical structure
(i.e., they are ergodic). We do not know a priori which is to be used, but
once the sequence starts in a given pure component Li it continues indefi-
nitely according to the statistical structure of that component.
As an example one may take two of the processes defined above and
assume pi = .2 and p^ = .8. A sequence from the mixed source
Lt ^^ tl, Lx \ .o Lt2
would be obtained by choosing first Zi or L^ with probabilities .2 and .8
and after this choice generating a sequence from whichever was chosen.
Except when the contrary is stated we shall assume a source to be ergodic.
This assumption enables one to identify averages along a sequence with
averages over the ensemble of possible sequences (the probability of a dis-
crepancy being zero). For example the relative frequency of the letter A
in a particular infinite sequence will be, with probability one, equal to its
relative frequency in the ensemble of sequences.
If Pi is the probability of state i and pi(]) the transition probability to
state jy then for the process to be stationary it is clear that the Pi must
satisfy equilibrium conditions:
Pi = l^PipiiD-
In the ergodic case it can be shown that with any starting conditions the
probabilities Pj(N) of being in state 7 after N symbols, approach the equi-
librium values as A^ ^ 00 .
6. Choice, Uncertainty and Entropy
We have represented a discrete information source as a Markoff process.
Can we define a quantity which will measure, in some sense, how much in-
formation is ''produced" by such a process, or better, at what rate informa-
tion is produced?
Suppose we have a set of possible events whose probabilities of occurrence
are /»i , ^2 , * • • , pn . These probabilities are known but that is all we know
concerning which event will occur. Can we find a measure of how much
"choice" is involved in the selection of the event or of how uncertain we are
of the outcome?
If there is such a measure, say Hipi , p2 j • • • , pn), it is reasonable to re-
quire of it the following properties:
1. H should be continuous in the />, .
1 . .
2. If all the pi are equal, pi = ~ , then // should be a monotonic increasing
n
MATHEMATICAL THEORY OF COMMVMfCATION 393
function of ;;. With equally likely events there is more choice, or un-
certainty, when there are more possible events.
3. If a choice be broken down into two successive choices, the original
H should be the weighted sum of the individual values of H, The
meaning of this is illustrated in Fig. 6. At the left we have three
possibilities pi = h p2 = i, P^ = h ^^ the right we first choose be-
tween two possibilities each with probability §, and if the second occurs
make another choice with probabilities |, J. The final results have
the same probabilities as before. We require, in this special case,
that
^(zj 3> e) ~ " (a? 2/ I 2^(3> 3/
The coefficient § is because this second choice only occurs half the time.
}/3
1/6
Fig. 6 — Decomposition of a choice from three possibilities.
In Appendix II, the following result is established:
Theorem 2: The only H satisfying the three above assumptions is of the
form:
n
H = -KJ2 pi log pi
1=1
where K is a. positive constant.
This theorem, and the assumptions required for its proof, are in no way
necessary for the present theory. It is given chiefly to lend a certain plausi-
bilitv to some of our later definitions. The real iustification of these defi-
nitions, however, will reside in their implications.
Quantities of the form// ^ —2 pi log pi (the constant A' merely amounts
to a choice of a unit of measure) play a central role in information theory as
measures of information, choice and uncertainty. The form of H will be
recognized as that of entropy as defined in certain formulations of statistical
mechanics^ where pi is the probability of a system being in cell i of its pha^e
space. H is then, for example, the H in Boltzmann's famous H theorem.
We shall call // = — "E pi log pi the entropy of the set of probabilities
^ See, for example, R. C. Tolman, "Principles of Statistical Mechanics," Oxford,
Clarendon, 1938.
394
BELL SYSTEM TECHNICAL JOURNAL
pit ' " t Pn * If a; is a chance variable we will write 11 (x) for its entropy;
thus X is not an argument of a function but a label for a number, to differen-
tiate it from H(y) say, the entropy of the chance variable y.
The entropy in the case of two possibilities with probabilities p and q =
1 — p) namely
H = -{:p\ogp-\- qlogq)
is plotted in Fig. 7 as a function of ^.
The quantity H has a number of interesting properties which further sub-
stantiate it as a reasonable measure of choice or information.
1.0
.9
.a
.7
H .6
BITS
.5
.4
.2
1
/^
"^
\
/
r 1
\
\
»
j
/
\
V
.
/
\
/
1
\
j
1
\
I
'
'
\
1
\
/
•
1 ■
\
!
J
/
1
\
.1
.2 .3
,5
P
6
.7
6
9
LO
Fig. 7 — Entropy in the case of two possibilities with probabilities p and (1 — p),
1. U = if and only if all the Pi but one are zero, this one having the
value unity. Thus only when we are certain of the outcome does H vanish.
Otherwise // is positive.
2. For a given «, fiT is a maximum and equal to log n when all the pi are
equal
I i.e.* I •
\ nj
This is also intuitively the most uncertain situation.
3. Suppose there are two events, x and y, in question with m possibilities
for the first and n for the second. Let p{%j) be the probability of the joint
occurrence of i for the first and _; for the second. The entropy of the joint
event is
U{x, y) = - S pih J) log p{i, j)
. w m
MATHEMATICAL THEORY OF COMMUNICATION 395
while
H{x) = - E P(h J) log E P(h j)
tt$
H{y) = - T. pa, j) \oslL P{i, i) .
It is easily shown that
H(x, y) < H(x) + H{y)
with equality only if the events are independent (i.e., p{ij j) — p{i) p(j)).
The uncertainty of a joint event is less than or equal to the sum of the
individual uncertainties.
4. Any change toward equalization of the probabilities pi j p2 y ■ " j pn
increases H. Thus if pi < p2 and we increase pi , decreasing Pi an equal
amount so that pi and p2 are more nearly equal, then H increases. More
*
generally, if we perform any ''averaging" operation on the pi of the form
pi = Zj <^i3p3
I
where E ^^v ~ E ^ti = Ij s-^^d all aij > 0, then H increases (except in the
special case where this transformation amounts to no more than a permuta-
tion of the pj with H of course remaining the same).
5. Suppose there are two chance events x and y as in 3, not necessarily
independent. For any particular value i that x can assume there is a con-
ditional probability pi(j) that y has the value _;. This is given by
^..s Pih j)
^'^^^ = E P(i. j) '
3
We define the conditional entropy of y, H x(y) as the average of the entropy
of y for each value of x, weighted according to the probability of getting
that particular x. That is
^xiy) = -Yl p{i, j) log pi{j),
* -i —
This quantity measures how uncertain we are of y on the average when we
know X, Substituting the value of pi(J) we obtain
Hxiy) = -E Pi^j i) log Pih j) + E P(h j) log E Pih j)
= H{x, y) - F(jt:)
or
H(x, y) = ZTCt) + H,{y)
m BELL SYSTEM TRCUmCAL JOURNAL
Th6 Uncertainty (or entropy) of the joint event Xj y is the uneertainty of x
plus the uncertainty of y when x is known.
6. From 3 and 5 we have
H{x) + H{y) > H(x, y) - H(x) + H ,{y)
Hence
H(y) > HXy)
The uncertainty of y is never increased by knowledge of x. It will be de-
creased unless X and y are independent events, in which case it is not changed.
7. The Entropy of an Information Source
Consider a discrete source of the finite state type considered above.
For each possible state i there will be a set of probabilities pi{j) of pro-
ducing the various possible symbols /. Thus there is an entropy Hi for
each state. The entropy of the source will be defined as the average of
these Hi weighted in accordance with the probability of occurrence of the
states in question:
^ = E Pi Hi
i
I
= - Z ^i Mi) log Mi)
».;
This is the entropy of the source per symbol of text. If the Markoff proc-
ess is proceeding at a definite time rate there is also an entropy per second
H' — 2^ fi Hi
»
where /t is the average frequency (occurrences per second) of state i. Clearly
H' = mH
ij
where m is the average number of symbols produced per second. H or 77'
measures the amount of information generated by the source per symbol
or per second. If the logarithmic base is 2, they will represent bits per
symbol or per second.
If successive symbols are mdependent then 77 is simply — S ^i log pi
where pi is the probability of symbol i. Suppose in this case we consider a
long message of A'' symbols. It will contain with high proljability about
piN occurrences of the first symbol, piN occurrences of the second, etc.
Hence the probability of this particular message will be roughly
p = pr"fP---pi:
or
MATHEMATICAL THEORY OF COMMUNICATION 397
log /> = TV 23 Pi log Pi
i
log p = -NH
N '
H is thus approximately the logarithm of the reciprocal probability of a
typical long sequence divided by the number of symbols in the sequence.
The same result holds for any source. Stated more precisely we have (see
Appendix III):
Theorem 3 : Given any e > and 6 > 0, we can find an iVo such that the se-
quences of any length N > Nq fall into two classes:
1, A set whose total probability is less than e.
2. The remainder, all of whose members have probabilities satisfying the
inequality
-1
log A _ f^
N
< 5
log p ^
In other words we are almost certain to have — ^f — very close to H when N
N
is large.
A closely related result deals with the number of sequences of various
probabilities. Consider again the sequences of length N and let them be
arranged in order of decreasing probability. We define n(q) to be the
number we must take from this set starting with the most probable one in
order to accumulate a total probability q for those taken.
Theorem 4 :
•at -00 iV
when q does not equal cfi: 1.
We may interpret log w(^) as the number of bits required to specify the
sequence when we consider only the most probable sequences with a total
probability q. Then — ^-rp^ is the number of bits per symbol for the
specification. The theorem says that for large N this will be independent of
q and equal to H. The rate of growth of the logarithm of the number of
reasonably probable sequences is given by Hj regardless of our interpreta-
tion of ^'reasonably probable." Due to these results, which are proved in
appendix III, it is possible for most purposes to treat the long sequences as
though there were just 2"^ of them, each with a probability 2"^^^.
398 BELL SYSTEM TECHNICAL JOURNAL
The next two theorems show that H and W can be determined by limit-
ing operations directly from the statistics of the message sequences, without
reference to the states and transition probabilities between states.
Theorem 5: Let p{Bi) be the probability of a sequence Bi of symbols from
the source. Let
. On = -^T^p(Bi) log p{Bi)
where the sum is over all sequences Bi containing N symbols. Then Gx
is a monotonic decreasing function of N and
Lim Gn = H,
N-*ao
Theorem 6: Let piBi, Sj) be the probability of sequence Bi followed by
, symbol Sj and pBi(Sj) = p{Bi y Sj)/p{Bi) be the conditional probability of
^j after B i . Let
+
Fn= -J2 p(Bi, Sj) log Ps, (Sj)
where the sum is over all blocks Bi oi N ~ 1 symbols and over all symbols
Sj . Then Fjt is a monotonic decreasing function of N,
r
Fy = NG^-{N - 1)0^-1,
Fs S Gif ,
and Lim Fn = H*
These results are derived in appendix III. They show that a series of
approximations to H can be obtained by considering only the statistical
structure of the sequences extending over 1, 2, • • • N symbols. Fn is the
better approximation. In fact Fk is the entropy of the N order approxi-
mation to the source of the type discussed above. If there are no statistical
influences extending over more than N symbols, that is if the conditional
probability of the next symbol knowing the preceding {N — 1) is not
changed by a knowledge of any before that, then Fn = H. Fjf of course is
the conrlitioTial entropy of the next symbol when the (iV — 1) preceding
ones arc known, while Gjv is the entropy per symbol of blocks of N symbols.
The ratio of the entropy of a source to the maximum value it could have
while still restricted to the same symbols will be called its relative entropy.
This is the maximum compressron possible when we encode into the same
One minus the relative entropy is the redundancy. The rcduu-
MATHEMATICAL THEORY OF COMMUNICATION 399
dancy of ordinary English, not considering statistical structure over greater
distances than about eight letters is roughly 50%. This means that when
we write English half of what we write is determined by the structure of the
language and half is chosen freely. The figure 50% was found by several
independent methods which all gave results in this neighborhood. One is
by calculation of the entropy of the approximations to English. A second
method is to delete a certain fraction of the letters from a sample of English
text and then let someone attempt to restore them. If they can be re-
stored when 50% are deleted the redundancy must be greater than 50%.
A third method depends on certain known results in cryptography;
Two extremes of redundancy in English prose are represented by Basic
English and by James Joyces' book ''Finigans Wake." The Basic English
vocabulary is limited to 850 words and the redundancy is very high. This
is reflected in the expansion that occurs when a passage is translated into
Basic English. Joyce on the other hand enlarges the vocabulary and is
alleged to achieve a compression of semantic content.
The redundancy of a language is related to the existence of crossword
puzzles. If the redundancy is zero any sequence of letters is a reasonable
text in the language and any two dimensional array of letters forms a cross-
word puzzle. If the redundancy is too high the language imposes too
many constraints for large crossword puzzles to be possible. A more de-
tailed analysis shows that if we assume the constraints imposed by the
language are of a rather chaotic and random nature, large crossword puzzles
are just possible when the redundancy is 50%. If the redundancy is 33%j
three dimensional crossword puzzles should be possible, etc.
8. Representation op the Encoding and Decoding Operations
We have yet to represent mathematically the operations performed by
the transmitter and receiver in encoding and decoding the information.
Either of these will be called a discrete transducer. The input to the
transducer is a sequence of input symbols and its output a sequence of out-
put symbols. The transducer may have an internal memory so that its
output depends not only on the present input S3Tnbol but also on the past
history. We assume that the internal memory is finite, i.e. there exists
a finite number m of possible states of the transducer and that its output is
a function of the present state and the present input symbol. The next
state will be a second function of these two quantities. Thus a transducer
can be described by two functions:
400 BELL SYSTEM TECHNICAL JOURNAL
where: Xn is the m" input symbol,
fh
an is the state of the transducer when the n input symbol is introduced,
v„ is the output symbol (or sequence of output symbols) produced when
Xn is introduced if the state is a„.
If the output symbols of one transducer can be identified with the input
symbols of a second, they can be connected in tandem and the result is also
a transducer. If there exists a second transducer which operates on the out-
put of the first and recovers the original input, the first transducer will be
called non-singular and the second will be called its inverse.
Theorem 7: The output of a finite state transducer driven by a finite state
statistical source is a finite state statistical source, with entropy (per unit
time) less than or equal to that of the input. If the transducer is non-
singular they are equal.
■
Let a represent the state of the source, which produces a sequence of
symbols Xi ; and let jS be the state of the transducer, which produces, in its
output, blocks of symbols yj . The combined system can be represented
by the "product state space" of pairs (a, /3). Two points in the space,
(ai , jSi) and {a^ ^2), are connected by a line if ai can produce an x which
changes j8i to jSg , and this line is given the probability of that x in this case.
The line is labeled with the block of yj symbols produced by the transducer.
The entropy of the output can be calculated as the weighted sum over the
states. If we sum first on ^ each resulting term is less than or equal to the
corresponding term for a, hence the entropy is not increased. If the trans-
ducer is non-singular let its output be connected to the inverse transducer.
If Hi , H2 and Hz are the output entropies of the source, the first and
/ / f f
second transducers respectively, then Hi > H2 > H3 = Hi and therefore
III — "2 .
Suppose we have a system of constraints on possible sequences of the type
which can be represented by a linear graph as in Fig. 2. If probabilities
tf^ \ *
pij were assigned to the various lines connecting state i to state y this would
become a source. There is one particular assignment which maximizes the
resulting entropy fsee Appendix IV).
Theorem 8: Let the system of constraints considered as a channel have a
capacity C. If we assign
Pa ^ ^C w
where t\'f is the duration of the 5'* symbol leading from state i to state j
and the Bi satisfy
then H is maximized and equal to C.
MATHEMATtCAL THEORY OF COMMUNICATIOX 401
By proper assignment of the transition probabilities the entropy of sym-
bols on a channel can be maximized at the channel capacity.
9. T*TiE Fundamental Theorfm for a Noiseless Channel
We will now justify our interpretation of H as the rate of generating
information by proving that // determines the channel capacity required
with most efficient coding.
Theorem 9: Let a source have entropy H (bits per symbol) and a channel
have a capacit}" C (bits per second). Then it is possible to encode the output
C
of the source in such a wav as to transmit at the average rate — — e svmbols
II
per second over the channel where e is arbitrarily small. It is not possible
C
to transmit at an average rate greater than — .
11
Q
The converse part of the theorem, that — cannot be exceeded, ma}' be
proved by noting that the entropy of the channel input per second is equal
to that of the source, since the transmitter must be non-singular, and also
this entropy cannot exceed the channel capacity. Hence W < C and the
number of symbols per second = W /H < C/H,
The first part of the theorem will be proved in two different ways. The
first method is to consider the set of all sequences of .V symbols produced by
the source. For X large we can divide these into two groups, one containing
less than 2 ' members and the second containing less than 2 * members
(where R is the logarithm of the number of different symbols) and having a
total probability less than /x. As iV increases t; and ju approach zero. The
number of signals of duration T in the channel is greater than 2 with
$ small when T is large. If we choose
r = ~ + X A
then there will be a sufficient number of sequences of channel symbols for
the high probability group when N and T are sufficiently large (however
small X) and also some additional ones. The high probability group is
coded in an arbitrary one to one way into this set. The remaining sequences
are represented by larger sequences, starting and ending with one of the
sequences not used for the high probability group. This special sequence
acts as a start and stop signal for a different code. lu between a sufficient
time is allowed to give enough different sequences for all the low probability
messages. This will require
402 BELL S YSTEM TECHNICAL JOVRNA L
where <p is small. The mean rate of transmission in message symbols per
second will then be greater than
= >-«(? + ^) + Kl +
.^"'
C
As N increases 5, X and <p approach zero and the rate approaches — .
B.
Another method of performing this coding and proving the theorem can
be described as follows: Arrange the messages of length N in order of decreas-
mg probabiHty and suppose their probabilities s^ie Pi> P2 > pz , . .> pn ^
Let P. = 2; pi ; that is P. is the cumulative probability up to, but not
1
ladadiagj p, . We first encode into a binarj' system. The binary code for
message s is obtained by expanding P^ as a binary number. The expansion
is carried out to nta places, where nts is the integer satisfying:
1 1
loga — < w, < 1 + log2 —
pM pB
Thus the messages of high probability are represented by short codes and
those of low probability by long codes. From these inequalities we have
' <P.< '
-1 •
2»j, — ^ 2"'»
The code for P, will differ from all succeeding ones in one or more of its
1
m, places, since all the remaining P.- are at least — - larger and their binary
expansions therefore differ in the first m, places. Consequently all the codes
are different and it is possible to recover the message from its code. If the
channel sequences are not already sequences of binary digits, they can be
ascribed binary numbers in an arbitrary fashion and the binary code thus
translated into signals suitable for the channel.
The average number H' of binary digits used per symbol of original mes-
sage is easily estimated. We have
N
But,
ir(iog.i)^.<i2../.,<^z(i +
P
and therefore,
MATBEMATICAL THEORY OF COM.UUXICATJOX 403
~^Ps log p,<H' <~- up, log /►.
As N increases — Sp, log pg approaches E, the entropy of the source and H*
approaches H.
\Xq see from this that the inefficiency in coding, when only a finite dela)* of
1
A" symbols is used, need not be greater than — plus the difference between
the true entropy H and the entropy Gn calculated for sequences of length iV.
The per cent excess time needed over the ideal is therefore less than
• Gv 1
— + — - — 1
H EN
This method of encoding is substantially the same as one found inde-
pendently by R. M. Fano.* His method is to arrange the messages of length
T in order of decreasing probability. Divide this series into two groups of
as nearly equal probability as possible. If the message is in the first group
its first binary digit wiU be 0, otherwise 1 . The groups are similarly divided
into subsets of nearly equal probability- and the particular subset determines
the second binary digit. This process is continued until each subset contains
only one message. It is easily seen that apart from minor differences (gen-
erally in the last digit) this amounts to the same thing as the arithmetic
process described above.
10. Discussion
In order to obtain the maximum power transfer from a generator to a load
a transformer must in general be introduced so that the generator as seen
from the load has the load resistance. The situation here is roughly anal-
ogous. The transducer which does the encoding should match the source
to the channel in a statistical sense. The source as seen from the channel
through the transducer should have the same statistical structure as the
source which maximizes the entropy in the channel. The content of
Theorem 9 is that, although an exact match is not in general possible, we can
approximate it as closely as desired. The ratio of the actual rate of trans-
mission to the capacity C may be called the efficiency of the coding system.
This is of course equal to the ratio of the actual entropy of the channel
s>Tnbols to the maximum possible entropy.
In general, ideal or nearly ideal encoding requires a long delay in the
transmitter and receiver. In the noiseless case which we have been
considering, the main function of this delay is to allow reasonably good
• Technical Report No. 65, The Research Laboratory'' of Electronics, M. I. T.
404 ■ BELL SYSTEM TECHNICAL JOURNAL
matching of probabilities to corresponding lengths of sequences. With a
good code the logarithm of the reciprocal probability of a long message
»
must be proportional to the duration of the corresponding signal, in fact
T
must be small for all but a small fraction of the long messages.
If a source can produce only one particular message its entropy is zero,
and no channel is required. For example, a computing machine set up to
calculate the successive digits of x produces a definite sequence with no
chance element. No channel is required to transmit" this to another
point. One could construct a second machine to compute the same sequence
at the point. However, this may be impractical. In such a case we can
choose to ignore some or all of the statistical knowledge we have of the
source. We might consider the digits of tt to be a random sequence in that
we construct a system capable of sending any sequence of digits. In a
similar way we may choose to use some of our statistical knowledge of Eng-
lish in constructing a code, but not all of it. In such a case we consider the
source with the maximum entropy subject to the statistical conditions we
wish to retain. The entropy of this source determines the channel capacity
which is necessary and sufficient. In the x example the only information
retained is that all the digits are chosen from the set 0, 1, . . ., 9. In the
case of English one might wish to use the statistical saving possible due to
letter frequencies, but nothing else. The maximum entropy source is then
the first approximation to English and its entropy determines the required
channel capacity.
11, Examples
As a simple example of some of these results consider a source which
produces a sequence of letters chosen from among Ay B, C, D with prob-
abilities I, }, J, i, successive symbols being chosen independently. We
have
II = _(llogi+ 1 log i^- flog i)
= i bits per symbol.
Thus we can aj>proximate a coding system to encode messages from this
source into binary digits with an average of ^ binary digit per symbol.
In this case we can actually achieve the limiting value by the following code
(obtained by the method of the second proof of Theorem 9) :
MATHEMATICAL THEORY OF COMMUNICATION 405
A
B
10
C
110
D
111
The average number of binary digits used in encoding a sequence of N sym-
bols will be
- iV(J X 1 + i X 2 + I X 3) = liV
It is easily seen that the binary digits 0, 1 have probabilities |, | so the H for
the coded sequences is one bit per symbol. Since, on the average, we have
binary symbols per original letter, the entropies on a time basis are the
same. The maximum possible entropy for the original set is log 4 = 2,
occurring when A^B.C^D have probabilities J, J, J, J. Hence the relative
entropy is f . We can translate the binary sequences into the original set of
symbols on a two-to-one basis by the following table:
7
4
00
A'
01
B'
10
a
11
D'
This double process then encodes the original message into the same symbols
but with an average compression ratio | .
As a second example consider a source which produces a sequence of ^'s
and 5's with probability p for A and q for B, li p < < q we have
: //= -log p''{\-py^'
= -p\ogp{\-pr-''^'
= p\og-
p
In such a case one can construct a fairly good coding of the message on a
0, 1 channel by sending a special sequence, say 0000, for the infrequent
symbol A and then a sequence indicating the number of 5*s following it.
This could be indicated by the binary representation with all numbers con-
taining the special sequence deleted. All numbers up to 16 are represented
as usual; 16 is represented by the next binary number after 16 which does
not contain four zeros, namely 17 = 10001, etc.
It can be shown that as /^ — > the coding approaches ideal provided the
length of the special sequence is properly adjusted.
406 BELL SYSTEM TECHNICAL JOURNAL
PART II: THE DISCRETE CHANNEL WITH NOISE
11. Representation of a Noisy Discrete Channel
We now consider the case where the signal is perturbed by noise during
transmission or at one or the other of the terminals. This means that the
received signal is not necessarily the same as that sent out by the trans-
mitter. Two cases may be distinguished. If a particular transmitted signal
always produces the same received signal, i.e. the received signal is a definite
function of the transmitted signal, then the effect may be called distortion.
If this function has an inverse — no two transmitted signals producing the
same received signal — distortion may be corrected, at least in principle, by
merely performing the inverse functional operation on the received signal.
The case of interest here is that in which the signal does not always undergo
the same change in transmission. In this case we may assume the received
signal £ to be a function of the transmitted signal S and a second variable,
the noise N,
E = f(S, N)
The noise is considered to be a chance variable just as the message was
above. In general it may be represented by a suitable stochastic process.
The most general type of noisy discrete channel we shall consider is a general-
ization of the finite state noise free channel described previously. We
assume a finite number of states and a set of probabilities
This is the probability, if the channel is in state a and symbol i is trans-
mitted, that symbol J win be received and the channel left in state jS. Thus
a and j3 range over the possible states, i over the possible transmitted signals
and j over the possible received signals. In the case where successive sym-
bols are independently perturbed by the noise there is only one state, and
the channel is described by the set of transition probabilities pi(j)f the prob-
ability of transmitted symbol i being received as j.
If a noisy channel is fed by a source there are two statistical processes at
work: the source and the noise. Thus there are a number of entropies that
can be calculated. First there is the entropy II{x) of the source or of the
input to the channel (these will be equal if the transmitter is non-singular).
The entropy of the output of the channel, i.e. the received signal, will be
denoted by 11 iy). In the noiseless case H(y) — H(x). The joint entropy of
input and output will be H(xy). Finally there are two conditional entro-
pies IIx(y) and Hy(x), the entropy of the output when the input is known
and conversely. Among these quantities we have the relations
H{x, y) = H(x) + H,(y) = H{y) + Hy(x)
MATHEMATICAL THEORY OF COMMUNICATION 407
All of these entropies can be measured on a per-second or a per-symbol
basis.
12. Equivocation and Channel Capacity
If the channel is noisy it is not in general possible to reconstruct the orig-
inal message or the transmitted signal with certainty by any operation on the
received signal E. There are, however, ways of transmitting the information
which are optimal in combating noise. This is the problem which we now
consider.
Suppose there are two possible s3anbols and 1, and we are transmitting
at a rate of 1000 symbols per second with probabilities po = pi = i . Thus
our source is producing information at the rate of 1000 bits per second. Dur-
ing transmission the noise introduces errors so that, on the average, 1 in 100
is received incorrectly (a as 1, or 1 as 0). What is the rate of transmission
of information? Certainly less than 1000 bits per second since about 1%
of the received symbols are incorrect. Our first impulse might be to say the
rate is 990 bits per second, merely subtracting the expected number of errors.
This is not satisfactory since it fails to take into account the recipient's
lack of knowledge of where the errors occur. We may carry it to an extreme
case and suppose the noise so great that the received s3anbols are entirely
independent of the transmitted symbols. The probability of receiving 1 is
^ whatever was transmitted and similarly for 0. Then about half of the
received synfbols are correct due to chance alone, and we would be giving
the system credit for transmitting 500 bits per second while actually no
information is being transmitted at all. Equally "good" transmission
would be obtained by dispensing with the channel entirely and flipping a
coin at the receiving point.
Evidently the proper correction to apply to the amount of information
transmitted is the amount of this information which is missing in the re-
ceived signal, or alternatively the uncertainty when we have received a
signal of what was actually sent. From our previous discussion of entropy
as a measure of uncertainty it seems reasonable to use the conditional
entropy of the message, knowing the received signal, as a measure of this
missing information. This is indeed the proper definition, as we shall see
later. Following this idea the rate of actual transmission, R, would be ob-
tained by subtracting from the rate of production (i.e., the entropy of the
source) the average rate of conditional entropy.
R = H{x) - Ey{x)
■9
The conditional entropy Hy{x) will, for convenience, be called the equi-
vocation. It measures the average ambiguity of the received signal.
40S BELL SYSTEM TECHNICAL JOVRNAL
In the example considered above, if a is received the a postericri prob-
ability that a was transmitted is .99, and that a 1 was transmitted is
.01. These figures are reversed if a 1 is received. Hence
; Hy{x) = - f.99 log .99 + 0.01 log 0.01]
= .081 bits/symbol
or 81 bits per second. We may say that the system is transmitting at a rate
1000 — 81 = 919bits per second. In the extreme case where a is equally
likely to be received as a or 1 and similarly for 1, the a posteriori proba-
bilities are |, J and
Sy{x) = - fi log 1 + i log i]
= 1 bit per symbol
or 1000 bits per second. The rate of transmission is then as it should
be.
The following theorem gives a direct intuitive interpretation of the
equivocation and also serves to justify it as the unique appropriate measure.
We consider a communication system and an observer (or auxiliar>^ device)
who can see both what is sent and what is recovered (with erro:s
due to noise). This observer notes the errors in the recovered message and
transmits data to the receiving point over a "correction channel" to enable
the receiver to correct the errors. The situation is indicated schematically
in Fig. 8.
Theorem 10: If the correction channel has a capacity equal to Hy{x) it is
possible to so encode the correction data as to send it over this channel
and correct all but an arbitrarily small fraction e of the errors. This is not
possible if the channel capacity is less than Hy{x).
Roughly then, Hy{x) is the amount of additional information that must be
supplied per second at the receiving point to correct the received message.
To prove the first part, consider long sequences of received message M'
and corresponding original message M. There will be logarithmically
THy(x) of the Af' s which could reasonably have produced each M'. Thus
we have THy{x) binary digits to send each T seconds. This can be done
with e frcfiuency of errors on a channel of capacity ny{x).
The second part can be proved by noting, first, that for any discrete chance
variables :r, y, 2 •
Hyix^ z) > HJx)
r
*
The left-hand side can be expanded to give
Ilyiz) + lly^ix) > fljx)
IIy,{x) > JIy{x) - Uy{z) > Hy{x) - IF {z)
MATHEMATICAL THEORY OF COMMUNICATION
409
If we identify x as the output of the source, y as the received signal and s
as the signal sent over the correction channel, then the right-hand side is the
equivocation less the rate of transmission over the correction channel. If
the capacity of this channel is less than the equivocation the right-hand side
will be greater than zero and Byz{x) > 0. But this is the uncertainty of
what was sent, knowing both the received signal and the correction signal.
If this is greater than zero the frequency of errors cannot be arbitrarily
small.
Example'.
Suppose the errors occur at random in a sequence of binary digits: proba-
bility p that a digit is wrong and q = i — p that it is right. These errors
can be corrected if their position is known. Thus the correction channel
need only send information as to these positions. This amounts to trans-
CCRRECTION DATA
SOURCE
TRANSMITTER
RECEIVER
CORRECTING
DEVICE
Fig. 8 — Schematic diagram of a correction system.
mitting from a source which produces binary digits with probability p for
1 (correct) and q for (incorrect). This requires a channel of capacity
— \plogp+ q log q]
which is the equivocation of the original system.
. The rate of transmission 7? can be written in two other forms due to the
identities noted above. We have
R = H(x) - Hy{x)
- H{y) - E^iy)
= H{x) + H{y) - H{x, y).
The first defining expression has already been interpreted as the amount of
information sent less the uncertainty of what was sent. The second meas-
410 BELL SYSTEM TECHNICAL JOURNAL
ures the amount received less the part of this which is due to noise. The
third is the sum of the two amounts less the joint entropy and therefore in a
sense is the number of bits per second common to the two. Thus all three
expressions have a certain intuitive significance.
The capacity C of a noisy channel should be the maximum possible rate
of transmission, i.e., the rate when the source is properly matched to the
channel. We therefore define the channel capacity by
C = Max {E{x) - Hy{x))
where the maximum is with respect to all possible information sources used
as input to the channel. If the channel is noiseless, Hy{x) — 0. The defini-
tion is then equivalent to that already given for a noiseless channel since the
maximum entropy for the channel is its capacity,
13. The Fundamental Theorem for a Discrete Channel with
Noise
It may seem surprising that we should define a definite capacity C for
a nulsy channel since we can never send certain information in such a case.
It is clear, however, that by sending the information in a redundant form the
probability of errors can be reduced. For example, by repeating the
message many times and by a statistical study of the different received
versions of the message the probability of errors could be made very small.
One would expect, however, that to make this probability of errors approach
zero, the redundancy of the encoding must increase indefinitely, and the rate
of transmission therefore approach zero. This is by no means true. If it
were, there would not be a very well defined capacity, but only a capacity
for a given frequency of errors, or a given equivocation; the capacity going
down as the error requirements are made more stringent. Actually the
capacity C defined above has a very definite significance. It is possible
to send information at the rate C through the channel with as small a fre-
quency of errors or equivocation as desired by proper encoding. This state-
ment is not true for any rate greater than C. If an attempt is made to
transmit at a higher rate than C, say C + i?i , then there w ill necessarily
be an equivocation equal to a greater than the excess Ri . Nature takes
payment by requiring just that much uncertainty, so that we are not
actually getting any more than C through correctly.
The situation is indicated in Fig. 9. The rate of information into the
channel is plotted horizontally and the equivocation vertically. Any point
above the heavy line in the shaded region can be attained and those below
cannot. The points on the line cannot in general be attained, but there will
usually be two points on the line that can.
MATHEMATICAL THEORY OF COMMUNICATION
411
These results are the main justification for the definition of C and will
now be proved.
Theorem 11. Let a discrete channel have the capacity C and a discrete
source the entropy per second H. li H < C there exists a coding system
such that the output of the source can be transmitted over the channel with
an arbitrarily small frequency of errors (or an arbitrarily small equivocation).
li H > C it is possible to encode the source so that the equivocation is less
than ^ — C + € where e is arbitrarily small. There is no method of encod-
ing which gives an equivocation less than H — C.
The method of proving the first part of this theorem is not by exhibiting
a coding method having the desired properties, but by showing that such a
code must exist in a certain group of codes. In fact we will average the
frequency of errors over this group and show that this average can be made
less than e. If the average of a set of numbers is less than e there must
exist at least one in the set which is less than e. This will establish the
desired result.
H(X)
Fig. 9 — The equivocation possible for a given input entropy to a channel.
The capacity C of a noisy channel has been defined as
C — Max {H{x) — Hyix))
where x is the input and y the output. The maximization is over all sources
which might be used as input to the channel.
Let Sq be a source which achieves the maximum capacity C. If this
maximum is not actually achieved by any source let ^o be a source which
approximates to giving the maximum rate. Suppose 5o is used as input to
the channel. We consider the possible transmitted and received sequences
of a long duration T. The following will be true:
1. The transmitted sequences fall into two classes, a high probability group
with about 2^^^*^ members and the remaining sequences of small total
probability.
2. Similarly the received sequences have a high probability set of about
2^" ^ members and a low probability set of remaining sequences.
3. Each high probability output could be produced by about 2^^" •* inputs.
The probability of all other cases has a small total probability.
412 ' BELL SYSTEM TECHNICAL JOURNAL
All the e's and 6's implied by the words *'small" and *'about" in these
statements approach zero as we allow T to increase and Sq to approach the
maximizing source.
The situation is summarized in Fig. 10 where the input sequences are
points on the left and output sequences points on the right. The fan of
cross lines represents the range of possible causes for a typical output.
Now suppose we have another source producing information at rate R
with R < C. In the period T this source will have 2 high probability
outputs. We wish to associate these with a selection of the possible channe
M
2 H(y)T
HIGH PROBABILITY, -~l»o u/ i-r
MESSAGES _- -:=====°^ 2^v^'
. , • HIGH PROBABILITY
hJ'XJT RECEIVED SIGNALS
REASONABLE CAUSES
FOR EACH E
• REASONABLE EFFECTS •
FROM EACH M
Fig. 10— Schematic representation of the relations between inputs and outputs in a
channel.
inputs in such a way as to get a small frequency of errors. We will set up
this association in all possible ways (using, however, only the high proba-
bility group of inputs as determined by the source So) and average the fre-
quency of errors for this large class of possible coding systems. This is the
same as calculating the frequency of errors for a random association of the
messages and channel inputs of duration T. Suppose a particular output
yi is observed. What is the probability of more than one message in the set
of possible causes of yi? There are 2™ messages distributed at random in
2 points. The probability of a particular point being a message is
2
TiR—H («) )
MATHEMATICAL THEORY OF COMMUNICATION 413
The probability that none of the points in the fan is a message (apart from
the actual originating message) is
p _ r-« 2'^^^~"^^^h2'^^v^^^
Now R < H{x) — Hy{x) so R ~ n(x) = — //„(a:) — 17 with rj positive-
Consequently
p = \i — 2~^^v^'^^~^^p'^^«^^^
approaches (as T" — > 00 )
1 - 2
-Tn
Hence the probabihty of an error approaches zero and the first part of the
theorem is proved.
The second part of the theorem is easily shown by noting that we could
merely send C bits per second from the source, completely neglecting the
remainder of the information generated. At the receiver the neglected part
gives an equivocation H{x) — C and the part transmitted need only add e.
This limit can also be attained in many other ways, as will be shown when we
consider the continuous case.
The last statement of the theorem is a simple consequence of our definition
of C. Suppose we can encode a source with R = C -\- ain such a way as to
obtain an equivocation Hy{x) = a — e with € positive. Then R = H{x) =
C + a a
n{x) — Hy{x) = C + c
with e positive. This contradicts the definition of C as the maximum of
H{x) — Hy{x).
Actually more has been proved than was stated in the theorem. If the
average of a set of numbers is within e of their maximum, a fraction of at
most V€ can be more than v € below the maximum. Since e is arbitrarily
small we can say that almost all the systems are arbitrarily close to the ideal.
14. Discussion
The demonstration of theorem 11, while not a pure existence proof, has
some of the deficiencies of such proofs. An attempt to obtain a good
approximation to ideal coding by following the method of the proof is gen-
erally impractical. In fact, apart from some rather trivial cases and
certain limiting situations, no explicit description of a series of approxima-
tion to the ideal has been found. Probably this is no accident but is related
to the difficulty of giving an explicit construction for a good approximation
to a random sequence.
414 BELL SYSTEM TECHNICAL JOURNAL
An approximation to the ideal would have the property that if the signal
is altered in a reasonable way by the noise, the original can still be recovered.
In other words the alteration will not in general bring it closer to another
reasonable signal than the original. This is accomplished at the cost of a
certain amount of redundancy in the coding. The redundancy must be
introduced in the proper way to combat the particular noise structure
involved. However, any redundancy in the source will usually help if it is
utilized at the receiving point. In particular, if the source already has a
certain redundancy and no attempt is made to eliminate it in matching to the
channel, this redundancy will help combat noise. For example, in a noiseless
telegraph channel one could save about 50% in time by proper encoding of
the messages. This is not done and most of the redundnacy of English
remains in the channel symbols. This has the advantage, however, of
allowing considerable noise in the channel. A sizable fraction of the letters
can be received incorrectly and still reconstructed by the context. In
fact this is probably not a bad approximation to the ideal in many cases,
since the statistical structure of English is rather involved and the reasonable
English sequences are not too far (in the sense required for theorem) from a
random selection.
As in the noiseless case a delay is generally required to approach the ideal
encoding. It now has the additional function of allowing a large sample of
noise to affect the signal before any judgment is made at the receiving point
as to the original message. Increasing the sample size always sharpens the
possible statistical assertions.
The content of theorem 11 and its proof can be formulated in a somewhat
different way which exhibits the connection wnth the noiseless case more
clearly. Consider the possible signals of duration T and suppose a subset
of them is selected to be used. Let those in the subset all be used with equal
probability, and suppose the receiver is constructed to select, as the original
signal, the most probable cause from the subset, when a perturbed signal
is received. - We define N{T, q) to be the maximum number of signals we
can choose for the subset such that the probability of an incorrect inter-
pretation is less than or 6qual to ^.
log NiT a)
Theorem 12: Lim — ^ — 2,—^-^ = C, where C is the channel capacity, ])ro-
7-*ao i
vided that q does not equal or 1 .
In other words, no matter how we set our limits of reliability, we can
distinguish reliably in time T enough messages to correspond to about CT
bits, when T is sufficiently large. Theorem 12 can be compared with the
definition of the capacity of a noiseless channel given in section 1.
MATUEMATICAL THEORY OF COMMUNICATION 415
15. ExAiiPLE OF A Discrete Channel and Its Capacity
A simple example of a discrete channel is indicated in Fig. 11. There
are three possible symbols. The first is never affected by noise. The second
and third each have probability p of coming through undisturbed, and q
of being changed into the other of the pair. We have (letting a — — [p log
— >
m
q^
v^
^^
.
TRANSMITTED ^V>^ RECEIVED
SYMBOLS J^^^ SYMBOLS
P
Fig. 11 — Example of a discrete channel.
p -{• q\og (j\ and P and Q be the probabihties of using the first or second
symbols)
Hix) = -P log P - 2Q log Q
EM = 2Qa
We wish to choose P and Q in such a way as to maximize H{x) — Hy(x)y
subject to the constraint P -\- 2Q = 1. Hence we consider
U = -P log P - 2Q\ogQ- 2Qa + X(P + 2Q)
dU
^ = -1 - logP + X =
dP ^
dlJ
— = -2 -2 log Q - 2a + 2X = 0.
m m
ing X
log P = log Q -\- a
/3+ 2 ^ /3 + 2*
The channel capacity is then
C = log — ^
416 BELL SYSTEM TECHNICAL JOURNAL
Note how this checks the obvious values in the cases p — ^ and p ~ \ .
In the first, jS = 1 and C ~ log 3, which is correct since the channel is then
noiseless with three possible symbols. If /? = 5, /3 = 2 and C = log 2.
Here the second and third symbols cannot be distinguished at all and act
together like one symbol. The first S3anbol is used with probability P =
^ and the second and third together with probability J . This may be
distributed in any desired way and still achieve the maximum capacity.
For intermediate values of p the channel capacity will He between log
2 and log 3. The distinction between the second and third symbols conveys
some information but not as much as in the noiseless case. The first symbol
is used somewhat more frequently than the other two because of its freedom
from noise.
16. The Chankel Capacity in Certain Special Cases
If the noise affects successive channel symbols independently it can be
described by a set of transition probabilities pij . This is the probability,
if symbol i is sent, that j will be received. The maximum channel rate is
then given by the maximum of
2I Pi pij log Zl Pi pij — zl Pi pij log pij
Mm V ■ ■■
where we vary the Pi subject to SP^ = 1. This leads by the method of
Lagrange to the equations,
P^i
£ ^«i log V P A ^ ^ ^ = 1» 2,
* *
Multiplying by P, and summing on s shows that ^ = —C. Let the inverse
of p,j (if it exists) be hgt so that ^ hgtpaj = 5ij . Then;
s
ZL ^Bt p,3 log Psj — log ZZ Pi pit ~ —C ZL fht .
s.i i »
Hence :
Zl Pi pit = exp [C ^hst -\- zl hst psj log Psj]
«.;
or,
Pi = 2Z hit exp [C ^ h^t + ^ KtPBj log p8}\-
This is the system of equations for determining the maximizing values of
Pi , with C to be determined so that Z Pi — 1. When this is done C will be
the channel capacity, and the P» the proper probabilities for the channel
symbols to achieve this capacity.
MATHEMATICAL THEORY OF COMMUNICATION
417
If each input symbol has the same set of probabilities on the lines emerging
from it, and the same is true of each output symbol, the capacity can be
easily calculated. Examples are shown in Fig. 12. In such a case Hx(y)
is independent of the distribution of probabilities on the input symbols, and
is given by —^ pi log pi where the pi are the values of the transition proba-
bilities from any input symbol. The channel capacity is
Max [H(y)
H.m
= M3ixH(y) + I^pilogpi.
The maximum of H{y) is clearly log m where m is the number of output
V
\></
^
^\/\
>
>o<^
"y.
--^ ^\\
a b c
Fig- 12 — Examples of discrete channels with the same transition probabilities for each
input and for each output.
symbols, since it is possible to make them all equally probable by making
the input symbols equally probable. The channel capacity is therefore
C = \ogm-\- ^ pi log pi
In Fig. 12a it would be
C = log 4 - log 2 = log 2.
This could be achieved by using only the 1st and 3d symbols. In Fig. 12b
C = log 4 - I log 3 - I log 6
= log 4 — log 3 - J log 2
= logi2^
In Fig. 12c we have
C = log 3 - I log 2 - ^ log 3 - J log 6
= log
3
2*3^6^ ■
418 BELL SYSTEM TECHNICAL JOURNAL
Suppose the symbols fall into several groups such that the noise never
causes a symbol in one group to be mistaken for a symbol in another group.
Let the capacity for the wth group be Cn when we use only the symbols
in this group. Then it is easily shown that, for best use of the entire set,
the total probability Pn of all symbols in the nth. group should be
■* n
2^n
22^
Within a group the probability is distributed just as It would be if these
were the only symbols being used. The channel capacity is
C = log 22"^".
17. An Example of Efficient Coding
The following example, although somewhat unrealistic, is -a case in which
exact matching to a noisy channel is possible. There are two channel
symbols, and 1, and the noise affects them in blocks of seven symbols. A
block of seven is either transmitted >Yithout error, or exactly one symbol of
the seven is incorrect. These eight possibilities are equally likely. We have
C = Max lH(y) - H,{y)]
= II7 + flogi]
= y bits/symbol .
An efficient code, allowing complete correction of errors and transmitting at
the rate C, is the following (found by a method due to R. Hamming) :
Let a block of seven symbols be Xi, Xzy . . • X7. Of these X3, Xb, X^ and
X7 are message symbols and chosen arbitrarily by the source. The other
three are redundant and calculated as follows:
X4 is chosen to make a — X4 + Xs -j- Xe + X7 even
X2 " ** " " /3 = Jta + X3 + Xe + X, "
Xi " " " 'f 7 = Xi + ^3 + :^6 + Xj "
When a block of seven is received a, /3 and 7 are calculated and if even called
zero, if odd called one. The binary number « /3 7 then gives the subscript
of the Xi thai is incorrect (if there was no error).
APPENDIX 1
TiTE Growth of the Number of Blocks of Symbols With A
Finite State Condition
Let Ni{L) be the number of blocks of symbols of length L ending in state
i. Then we have
%M
MATHEMATICAL THEORY OF COXfMUNICATTON 419
where b^ , bij , . . , bij are the length of the symbols which may be chosen
in state i and lead to state j. These are linear difference equations and the
behavior as Z, ^ « must be of the type
Substituting in the difference equation
AjW- = Y^AiW^-^Yi
or
Ai = T.AiW'^Yi
E (Z w^"^ - 5,,)^, =
For this to be possible the determinant
Z^diO = Ia,y| = IZ W^-*'^'-* - 5,,
must vanish and this determines IF, which is, of course, the largest real root
of Z) = 0.
The quantity C is then given by .
. XogXAjW^
C = Lim 'z^^.^;:^!^ = jo2 w
L—*QQ
L '"""
and we also note that the same growth properties result if we require that all
blocks start in the same (arbitrarily chosen) state.
APPENDIX 2
Derivation 0¥ H = —X pi log pi
(11 1 \
- 1 - 1 ' ' ' i -] — A(n). From condition (3) we can decompose
n n n)
a choice from s^ equally likely possibilities into a series of m choices each
from 5 equally likely possibilities and obtain
A(f) = mA{s)
Similarly
A (/") ^ nA (I)
We can choose n arbitrarily large and lind an m to satisfy
s^ < f <s
im+i)
420
BELL SYSTEM TECHNICAL JOURNAL
Thus, taking logarithms and dividing by n log 5,
— <
n
log i ^ m , \
r^- < - + - or
log s n n
m
ft
log /
log 5
< €
where e is arbitrarily small.
Now from the monotonic property of A (n)
A(s^) < A(f) < ^(^^0
fnA{s) < nA (t) < {fn+ 1) A {s)
Hence, dividing by nA{s),
m ^ A{t) m 1
n A{s) n n
or
m
n
A{i)
A(s)
< €
A{t) _ log t
A{s) log 5
< 26 Ait) == -^log;
where K must be positive to satisfy (2) .
Now suppose we have a choice from n possibilities with commeasurable prob-
abilities pi -
2) no-
where the Wt are integers. We can break down a choice
from Xtti possibilities into a choice from n possibilities with probabilities
pi. , , pn and then, if the ith was chosen, a choice from w* with equal prob-
abilities. Using condition 3 again, we equate the total choice frnm Sw*
as computed by two methods
K log S«i =
I , . . . , pv) -\- K'Z pi log fii
Hence
H = K\Z piXog^fii — i: pi log «i]
= -K-^PiXo^
Hi
~ —KXpilogpi,
If the pi are incommeasurable, they may be approximated by rationals and
the same expression must hold by our continuity assumption. Thus the
expression holds in general. The choice of coefficient iC is a matter of con-
venience and amounts to the choice of a unit of measure.
APPENDIX 3
Theorems on Ehgodic Sources
If it is possible to go from any state with P > to any other along a path
of probability ^ > 0, the system is ergodic and the strong law of large num-
bers can be applied. Thus the number of times a given path pij in the net-
MATHEMATICAL THEORY OF COMMUNICATION 421
i
work is traversed in a long sequence of length N is about proportional to the
probability of being at i and then choosing this path, PipuN, If N is large
enough the probability of percentage error ± 6 in this is less than € so that
for all but a set of small probability the actual numbers lie within the limits
{Pipij ± 5)N
Hence nearly all sequences have a probability p given by
p = ni^lj"*^*'^^^"^
and — ^ is limited by
iV
or
log p
^^t =. i:(p,pij ±d) log pij
XPi pa log pij
< V
This proves theorem 3.
Theorem 4 follows immediately from this on calculating upper and lower
bounds for n{q) based on the possible range of values of p in Theorem 3.
In the mixed (not ergodic) case if
L = ^ Pi Li
and the entropies of the components are Hi > H2 ^ . . . > ffn we have the
Theorem: Lim ^ ~ <p{q) is a decreasing step function,
8 — 1 9
<p(q) = Hs in the interval 2^ a» < 5 < 2^ a,- .
1 1
To prove theorems 5 and 6 first note that Fn is mono tonic decreasing be-
cause increasing N adds a subscript to a conditional entropy. A simple
substitution for pBi (Sj) in the definition of F^f shows that
Fy = N Gti - {N - 1) Gjv-i
1
and summing this for all N gives Gn = -rzl^ Fs . Hence Gn > Fjf and Gif
Jy
monotonic decreasing. Also they must approach the same limit. By using
theorem 3 we see that Lim Gn = H.
N-*eo
Maximizing the Rate for a System of Constr.4ints
Suppose we have a set of constraints on sequences of symbols that is of
the finite state type and can be represented therefore by a linear graph.
422 BELL SYSTEM TECHNICAL JOURNAL
Let Uj be the lengths of the various symbols that can occur in passing from
state /' to state j. What distribution of probabilities P* for the different
states and pYj for choosing symbol s in state i and going to state^ maximizes
the rate of generating information under these constraints? The constraints
define a discrete channel and the maximum rate must be less than or equal
to the capacity C of this channel, since if all blocks of large length were
equally likely, this rate would result, and if possible this would be best. We
will show that this rate can be achieved by proper choice of the Pi and pi) .
The rate in question is
^Pii^plf(\f m'
Let tii = Z 4?. Evidently for a maximum p^'] = k exp i^ . The con-
s
straints on maximization are XPi = 1, Z^ pij = 1,^ Pi{pij ~ hij) = 0.
-■«
}
Hence we maximize
U = ~'^^tj'l ^J ^'' + X Z Pi + 2m< Pij + 2%- Piipii - Sii)
9
dpij M^
Solving for pa
Pij ~~ A % -Dj U
Since
i i
The correct value of D is the capacity C and the Bj are solutions of
^ i = 2 BjC~^' '■
for then
^*' Bi
zPi^cr^'^ - p,
Bi
or
MATHEMATICAL THEORY OF COMMUNICATION
423
Bi Bi
So that if Xi satisfy
Both of the sets of equations for 5i and 71 can be satisfied since C is such that
In this case the rate is
sp.^.v log |; c-^"
2 Pi Pij <ij
ZPi pi j log
= C -
-By
^Pi Pij iij
but
^PipijiH Bj - log i^i) = Z Pi log 5y - ZPi log 2J, =
Hence the rate is C and as this could never be exceeded this is the maximum,
justifying the assumed solution.
{To be continued) *
A Mathematical Theory of Communication
By C. E. SHANNON
(Concluded from July 1948 issue)
*
PART III: MATHEMATICAL PRELIMINARIES
In this final installment of the paper we consider the case where the
signals or the messages or both are continuously variable, in contrast with
the discrete nature assumed until now. To a considerable extent the con-
tinuous case can be obtained through a limiting process from the discrete
case by dividing the continuum of messages and signals into a large but finite
number of small regions and calculating the various parameters involved on
a discrete basis. As the size of the regions is decreased these parameters in
general approach as limits the proper values for the continuous case. There
are, however, a few new effects that appear and also a general change of
emphasis in the direction of specialization of the general results to particu-
lar cases.
We will not attempt, in the continuous case, to obtain our results with
the greatest generality, or with the extreme rigor of pure mathematics, since
this would involve a great deal of abstract measure theory and would ob-
scure the main thread of the analysis. A preliminary ' study, however, indi-
cates that the theory can be formulated in a completely axiomatic and
rigorous manner which includes both the continuous and discrete cases and
many others. The occasional liberties taken with limiting processes in the
present analysis can be justified in all cases of practical interest.
18. Sets and Ensembles of Functions
We shall have to deal in the continuous case with sets of functions and
ensembles of functions. A set of functions, as the name implies, is merely a
class or collection of functions, generally of one variable, time. It can be
specified by giving an explicit representation of the various functions in the
set, or implicitly by giving a property which functions in the set possess and
others do not. Some examples are:
1. The set of functions:
fe(t) = sin (t + e).
Each particular value of 6 determines a particular function in the set.
623
624 BELL SYSTEM TECHNICAL JOURNAL
2. The set of all functions of time containing no frequencies over W cycles
per second.
3. The set of all functions limited in band to W and in amplitude to A .
4. The set of all English speech signals as functions of time.
An ensemble of functions is a set of functions together with a probability
measure whereby we may determine the probability of a function in the
set having certain properties.^ For example with the set,
/fi(0 = sin (/ + e),
we may give a probability distribution for 6^ P(6). The set then becomes
an ensemble.
Some further examples of ensembles of functions are:
1. A finite set of functions /^(O (A = 1, 2, • • * , w) with the probability of
fk being pk.
2. A finite dimensional family of functions
with a probability distribution for the parameters ai :
p(ai , • • ■ , a„)
For example we could consider the ensemble defined by
n
/fal , • • ■ , On , ^1 , • ♦ • y&n ; t) = zL «n sin uioit + dj
n = l
with the amplitudes a^ distributed normally and independently, and the
phrases 6i distributed uniformly (from to 2ir) and independently.
3. The ensemble
J./ ,x v^* sin irilWt — ft)
n^= — 00
with the Gi normal and independent all with the same standard deviation
\/N. This is a representation of ^^white'* noise, band-limited to the band
from to W cycles per second and with average power iV.'
^In maLhematical terminology the functions belong to a measure space whose total
measure is unity.
^ This representation can be used as a definition of band limited white noise. It has
certain advantages in that it involves fewer limiting operations than do definitions that
have been used in the past. The name **white noise,'' already firmly intrenched in the
literature, is perhaps somewhat unfortunate. In optics white light means either any
continuous spectrum as contrasted with a point spectrum, or a spectrum which is flat with
wavelenglh (which is not the same as a spectrum flat with frequency).
MATHEMATICAL THEORY OF COMMUNICATION 625
4. Let points be distributed on the i axis according to a Poisson distribu-
tion. At each selected point the function f{t) is placed and the different
functions added, giving the ensemble
00
•Z fit + k)
Jt=— 00
where the tk are the points of the Poisson distribution. This ensemble
can be considered as a type of impulse or shot noise where all the impulses
are identical.
5. The set of English speech functions with the probability measure given
by the frequency of occurrence in ordinary use.
An ensemble of functions fa{t) is stationary if the same ensemble results
when all functions are shifted any fixed amount in time. The ensemble
fe{t) = sin it ^6)
is stationary if 6 distributed uniformly from to 2ir. If we shift each func-
tion by ^1 we obtain
feit + h) = sin it -\- ti + 6)
= sin it + <p)
with (p distributed uniformly from to 2ir. Each function has changed
but the ensemble as a whole is invariant under the translation. The other
examples given above are also stationary.
An ensemble is ergodic if it is stationary, and there is no subset of the func-
tions in the set with a probability different from and 1 which is stationary.
The ensemble
sin it-\- e)
is ergodic. No subset of these functions of probability 5*^0, 1 is transformed
into itself under all time translations. On the other hand the ensemble
a sin (/ + 0)
with a distributed normally and 6 uniform is stationary but not ergodic.
The subset of these functions with a between and 1 for example is
stationary.
Of the examples given, 3 and 4 are ergodic, and 5 may perhaps be con-
sidered so. If an ensemble is ergodic we may say roughly that each func-
tion in the set is typical of the ensemble. More precisely it is known that
with an ergodic ensemble an average of any statistic over the ensemble is
equal (with probability 1) to an average over all the time translations of a
626 BELL SYSTEM TECHNICAL JOURNAL
particular function in the set. Roughly speaking, each function can be ex-
pected, as time progresses, to go through, with the proper frequency, all the
convolutions of any of the functions in the set.
Just as we may perform various operations on numbers or functions to
obtain new numbers or functions, we can perform operations on ensembles
to obtain new ensembles. Suppose, for example, we have an ensemble of
functions /«(/) and an operator T which gives for each function /«(/) a result
ga{t) :
g«(/) = TUit)
Probability measure is defined for the set ga{i) by means of that for the set
/„(/). The probability of a certain subset of the ga{i) functions is equal
to that of the subset of the/a(/) functions which produce members of the
given subset of g functions under the operation T. Physically this corre-
sponds to passing the ensemble through some device, for example, a filter,
a rectifier or a modulator. The output functions of the device form the
ensemble ga{t)'
A device or operator T will be called invariant if shifting the input merely
shifts the output, i.e., if
g«(0 = 7y.(o
implies
ga{.t + h) = Tfa{t + ^l)
for all fa{i) and all h . It is easily shown (see appendix 1) that if T is in-
variant and the input ensemble is stationary then the output ensemble is
stationary. Likewise if the input is ergodic the output will also be ergodic.
A filter or a rectifier is invariant under all time translations. The opera-
tion of modulation is not since the carrier phase gives a certain time struc-
ture. However, modulation is invariant under all translations which are
multiples of the period of the carrier.
Wiener has pointed out the intimate relation between the invariance of
physical devices under time translations and Fourier theory. He has
3 This is the famous ergodic theorem or rather one aspect of this theorem which \^as
proveti is somewhat different formulations by BirkhofF, von Neumann, and Koopman, and
subsequently generalized by Wiener, Hopf , Hurewicz and others. The literature on ergodic
theory is (luilf «'xtcnsive and the reader is referred to the papers of these writers for pre-
cise and general formulations; e.g., E. Hopf "Krgodentheorie" Ergebnisse der Mathematic
und ihrer Grenzgebiete, Vol. 5, "On Causality Statistics and Prol)ahility" Journal of
Mathematics and Physics, Vol. XIII, No. 1, 1934; N. Weiner 'The Krgodic Theorem"
Duke Mathematical Journal, Vol. 5, 1939.
* Communication theory is heavily indebted to Wiener for much of its basic philosophy
and theory. His classic NDRC report "The Interpolation, P2xtrapolation, and Smoothing
of Stationary Time Series," to appear soon in book form, contains the first clear-cut
formulation of communication theory as a statistical problem, the study of operations
MATHEMATICAL THEORY OF COMMUNICATION 627
shown, in fact, that if a device is linear as well as invariant Fourier analysis
is then the appropriate mathematical tool for dealing with the problem.
An ensemble of functions is the appropriate mathematical representation
of the messages produced by a continuous source (for example speech), of
the signals produced by a transmitter, and of the perturbing noise. Com-
munication theory is properly concerned, as has been emphasized by Wiener,
not with operations on particular functions, but with operations on en-
sembles of functions. A communication system is designed not for a par-
ticular speech function and still less for a sine wave, but for the ensemble of
speech functions.
19. Band Limited Eksembles of Functions
If a function of time /(/) is limited to the band from to TF cycles per
second it is completely determined by giving its ordinates at a series of dis-
1
Crete points spaced —r^ seconds apart in the manner indicated by the follow-
ing result.
Theorem 13: Let f{t) contain no frequencies over W.
Then
/ A _ Y^ „ sin 7r(2TF^ — n)
-qo T{2Wt — n)
where
^n — f
2W
In this expansion f(t) is represented as a sum of orthogonal functions.
The coefficients Xn of the various terms can be considered as coordinates in
an infinite dimensional "function space." In this space each function cor-
responds to precisely one point and each point to one function.
A function can be considered to be substantially limited to a time T if all
the ordinates Xn outside this interval of time are zero. In this case all but
2ni^ of the coordinates will be zero. Thus functions limited to a band IF
and duration T correspond to points in a space of 2T\V dimensions.
A subset of the functions of band W and duration T corresponds to a re-
gion in this space. For example, the functions whose total energy is less
on time series. This work, although chiefly concerned with the linear prediction and
filtering problem, is an important collateral reference in connection with the present paper.
We may also refer here to Wiener's forthcoming book "Cybernetics** dealing with the
general probjems of communication and control.
^ For a proof of this theorem and further discussion see the author's pajwr "Communi-
cation in the Presence of Noise" to be published in the Proceedings of the Institute of Radio
Engineers.
628 BELL SYSTEM TECHNICAL JOURNAL
than or equal to E correspond to points in a 2TW dimensional sphere with
radius r = \/2WE*
An eftsemble of functions of limited duration and band will be represented
by a probability distribution p{xi • • • Xn) in the corresponding n dimensional
space. If the ensemble is not limited in time we can consider the 2TW co-
ordinates in a given interval T to represent substantially the part of the
function in the interval T and the probability distribution p{xi , ■ • • , x^)
to give the statistical structure of the ensemble for intervals of that duration.
20. Entropy of a Continuous Distribution
The entropy of a discrete set of probabilities piy ■ • - pn has been defined as :
H = —X pi log pi •
-■
In an analogous manner we define the entropy of a continuous distribution
with the density distribution function p{x) by:
= — / p(x) log p{x) dx
With an n dimensional distribution p{xi , • ■ • , x„) we have
H = — \ • • • / p{xi • ' ' Xn) log p{xi , • ■ ■ , iCn) dXi " • • dXn .
I
If we have two arguments x and y (which may themselves be multi-dimen-
sional) the joint and conditional entropies of p{Xj y) are given by
r
A
H{x, y) = - J j Pi^y y) log Pi^f y) dx dy
and
B.{y) = -jf Pix, y) log ^^^ dx dy
Ey{x) = - // Pi^. y) log ^^^ dx dy
where
p{x) = j p{x, y) dy
ft
• p{y) = j p{x, y) dx.
§
The entropy of continuous distributions have most (but not all) of the
properties of the discrete case. In particular we have the following :
MATHEMATICAL THEORY OF COMMUNICATION 629
1. If ^c is limited to a certain volume v in its space, then H(x) is a maximum
and equal to log v when p{x) is constant I - J in the volume.
2. With any two variables x, y we have
H{x, y) < H{x) + H{y)
with equality if (and only if) x and y are independent, i.e., p{x^ y) == p{x)
piy) (apart possibly from a set of points of probability zero) .
3. Consider a generalized averaging operation of the following type:
p\y) = I ^(^» y)pi^) ^^
with
/ a{xj y) dx = I a{x, y) dy = 1, a(ic, y) > 0.
Then the entropy of the averaged distribution p'(y) is equal to or greater
than that of the original distribution p{x),
4. We have
H{x, y) = H{x) + Ihiy) = H{y) + Hy{x)
and
H.(y) < H(y) .
5. Let p{x) be a one-dimensional distribution. The form of p(x) giving a
maximum entropy subject to the condition that the standard deviation
of X be fixed at a is gaussian. To show this we must maximize
H{x) = — I pix) log p{x) dx
2
<7 =
/ p{oc)x dx and 1 = I p{x)
dx
as constraints. This requires, by the calculus of variations, maximizing
/ [—pM log p(x) + Xp(x)x^ + fxp(x)\ dx.
The condition for this is
— 1 — log p{x) -\~ \x + /I =
and consequently (adjusting the constants to satisfy the constraints)
630 BELL SYSTEM TECHNICAL JOURNAL
Similarly in n dimensions, suppose the second order moments of
p(xi , • • • , .r„) are fixed at Aij :
A
ij — I * ' * I jP^" ^j J^x^'l 3 * ' * J *^?i/ ^^1 • • • il>^^ ,
Then the maximum entropy occurs (by a similar calculation) when
p{xi , ' • ' , Xn) is the 71 dimensional gaussian distribution with the second
order moments Aij .
6. The entropy of a one-dimensional gaussian distribution whose standard
deviation is a is given by
H(x) = log y/2ir€(x.
This is calculated as follows:
I
-log p(x) =. log \/2ira- + --
H(x) = — I pM log p(x) dx
. = / Pi^) log V27r a dx + I p{x)
dx
2a
= log \/2Tra- +
2
2ff'
= log \/27r<r + log ve
= log \/2Tre(r.
Similarly the n dimensional gaussian distribution with
quadratic form aij is given by
la -I*
pixi , -• ,Xn) = k^2 ^^P ^ ~ 2 ^^*^' ^' ^'^
and the entropy can be calculated as
n/2
H - log (2tc)"'' I aij
i
where I an \ is the determinant whose elements are an .
7. If jc is limited to a half line {p{x) = for :«; < 0) and the first moment of
X is fixed at a:
a = / p(x)x dXy
mathe.ua tical theorv of COM.\frxrCA TION 631
then the maximum entropy occurs when
a
' and is equal to log ea.
8. There is one important difference between the continuous and discrete
entropies. In the discrete case the entropy measures in an absolute
way the randomness of the chance variable. In the continuous case the
measurement is relative to the coordinate system. If we change coordinates
the entropy will in general change. In fact if we change to coordinates
>'i • • • yn, the new entropy is given by
H{y) = j "I p(xi ■ ■ ■ Xn)J f - j log p{xi ■ ■ • Xn)J { jdyi
dyn
where J I - j is the Jacobian of the coordinate transformation. On ex-
panding the logarithm and changing variables to .ri • • • x„ , we obtain
= H(x) ~ I ' ' ' I Pi^i } ' ' ' ) ^n) log -^ ( - ) ^^1 ' * • ^^» •
Thus the new entropy is the old entropy less the expected logarithm of
the Jacobian. In the continuous case the entropy can be considered a
measure of randomness relative to an assumed standard, namely the co-
ordinate system chosen with each small volume element dxi • * • dxn given
equal weight. When we change the coordinate system the entropy in
the new system measures the randomness when equal volume elements
dyi • ' ■ dyn in the new system are given equal weight.
In spite of this dependence on the coordinate system the entropy
concept is as important in the continuous case as the discrete case. This
is due to the fact that the deriv^ed concepts of information rate and
channel capacity depend on the difference of two entropies and this
difference does not depend on the coordinate frame, each of the two terms
being changed by the same amount.
The entropy of a continuous distribution can be negative. The scale
of measurements sets an arbitrary zero corresponding to a uniform dis-
tribution over a unit volume. A distribution which is more confined than
this has less entropy and will be negative. The rates and capacities will,
however, always be non-negative.
9. A particular case of changing coordinates is the linear transformation
y.
J — / ^ ^ij ^i ■
t
632 BELL SYSTEM TECHNICAL JOURNAL
In this case the Jacobian is simply the determinant | aij \~^ and
H(y) = H(x) + log
C*^jj
t
In the case of a rotation of coordinates (or any measure preserving trans-
formation) / = 1 and//(y) = H{x).
21. Entropy of an Ensemble of Functions
Consider an ergodic ensemble of functions limited to a certain band of
width W cycles per second. Let
be the density distribution function for amplitudes :*:i " ■ oc^ at n successive
sample points. We define the entropy of the ensemble per degree of free-
dom by
W = — Lim - / • • • / p{xi ' • ' Xn) log p{xi , • - • , Xj^ dxi -- ^ dxn .
n^+oo ft J J
We may also define an entropy H per second by dividing, not by », but by
the time T in seconds for n samples. Since n = 2TWj H' = 2WH.
With white thermal noise p is gaussian and we have
H' = log y/lireN,
H ^W\og lireN.
For a given average power N, white noise has the maximum possible
entropy. This follows from the maximizing properties of the Gaussian
distribution noted above.
The entropy for a continuous stochastic process has many properties
analogous to that for discrete processes. In the discrete case the entropy
was related to the logarithm of the probability of long sequences, and to the
number of reasonably probable sequences of long length. In the continuous
case it is related in a similar fashion to the logarithm of the p'obabilily
density for a long series of samples, and the volume of reasonably high prob-
ability in the function space.
More precisely, if we assume p{xi • * • Xj^ continuous in all the Xi for all »,
then for sufficiently large «
\ogp
W
n
< e
for all choices of (jcj , • ■ • , Xn) apart from a set whose total probability is
less than 5, with 5 and e arbitrarily small. This follows from the ergodic
property if we divide the space into a large number of small cells.
MATHEMATICAL THEORY .OF COMMUNICATION 633
The relation of H to volume can be stated, as follows: Under the same as-
sumptions consider the n dimensional space corresponding to p{xi , ■ • • , :v„).
Let Vn{q} be the smallest volume in this space which includes in its interior
a total probability q. Then
Lim ^2^XM = W
n-*oo
n
provided q does not equal or 1.
These results show that for large n there is a rather well-defined volume (at
least in the logarithmic sense) of high probability, and that within this
volume the probability density is relatively uniform (again in the logarithmic
sense). '
In the white noise case the distribution function is given by
Since this depends only on Xxi the surfaces of equal probability density
are spheres and the entire distribution has spherical synunetry. The region
of high probability is a sphere of radius \/nN. As w — » oo the probability
of being outside a sphere of radius \^n{N + e) approaches zero and - times
the logarithm of the volume of the sphere approaches log \/2ireN-
In the continuous case it is convenient to work not with the entropy H of
an ensemble but with a derived quantity which we will call the entropy
power. This is defined as the power in a white noise limited to the same
band as the original ensemble and having the same entropy. In other words
if £?' is the entropy of an ensemble its entropy power is
Ni = J- exp 2H\
2-we
In the geometrical picture this amounts to measuring the high probability
volume by the squared radius of a sphere having the same volume. Since
white noise has the maximum entropy for a given power, the entropy power
of any noise is less than or equal to its actual power.
21. Entropy Loss in Linear Filters
Theorem 14: If an ensemble having an entropy H\ per degree of freedom
in band W is passed through a filter with characteristic F(/) the output
ensemble has an entropy
H,=H, + ^f^ log I F(/) f df.
634
BELL SYSTEM TECHNICAL JOURNAL
The operation of the filter is essentially a linear transformation of co-
ordinates. If we think of the different frequency components as the original
coordinate system, the new frequency components are merely the old ones
multiplied by factors. The coordinate transformation matrix is thus es-
TABLE I
GAiN
ENTROPY
POWER
FACTOR
ENTROPY
POWER GAIN
IN DECIBELS
IMPULSE RESPONSE
1-a;
t-Q/^ :
-8.68
siN^;rt
^^■^ifaAAiP^^^^^^P
(f)
-5.32
[ sin t
cost
1-6;^
0.364
-4.15
cos t - 1
cost ^ SIN t
2t
t^ J
vn^—
(iJ
-2.66
^ Jl (t)
2 +
I
2ff
-8.68 ff
cos o-dr)t-cost
a;
sentially diagonalized in terms of these coordinates. The Jacob ian of the
transformation is (for n sine and n cosine components)
n
-^ = II I n/d
i-.l
MATHEMATICAL THEORY OF COMMUNICATION 635
where the Ji are equally spaced through the band W, This becomes in
the limit
exp 1 j log I F(/) f df.
Since J is constant its average value is this same quantity and applying the
theorem on the change of entropy with a change of coordinates, the result
follows. We may also phrase it in terms of the entropy power. Thus if
the entropy power of the first ensemble is N\ that of the second is ■
Ni exp ^ j^ log I F(/) 1^ dj.
The final entropy power is the initial entropy power multiplied by the geo-
metric mean gain of the filter. If the gain is measured in db^ then the
output entropy power will be increased by the arithmetic mean dh gain
over W,
In Table I the entropy power loss has been calculated (and also expressed
in dh) for a number of ideal gain characteristics. The impulsive responses
of these filters are also given for W = 2ir, with phase assumed to be 0.
The entropy loss for many other cases can be obtained from these results.
1
For example the entropy power factor — for the first case also applies to any
gain characteristic obtained from 1 — w by a measure preserving transforma-
tion of the 0) axis. In particular a linearly increasing gain G(co) = to, or a
"saw tooth" characteristic between and 1 have the same entropy loss.
1
The reciprocal gain has the reciprocal factor. Thus - has the factor ^\
Raising the gain to any power raises the factor to this power.
22. Entropy of the Sxjm of Two Ensembles
If we have two ensembles of functions /«(/) and g^{t) we can form a new
ensemble by ''addition." Suppose the first ensemble has the probability
density function p{xi , - • • , ^n) and the second q{xi , • • • , Xn)' Then the
density function for the sum is given by the convolution:
r{xi , " ' ,Xn) = \ • " \ piyi y ■ ■ ■ ,yn)
• q(xi — yiy ■■ • jXn — yn) dyiydy2, • " jdyn.
Physically this corresponds to adding the noises or signals represented by
the original ensembles of functions.
636
BELL SYSTEM TECHNICAL JOURNAL
The following result is derived in Appendix 6.
Theorem 15: Let the average power of two ensembles be Ni and Ni and
let their entropy powers be Ni and Nz . Then the entropy power of the
sum, J^3 , is bounded by
Ni-\-N2<Nz<Ni + N2.
■m
White Gaussian noise has the peculiar property that it can absorb any
other noise or signal ensemble which may be added to it with a resultant
entropy power approximately equal to the sum of the white noise power and
the signal power (measured from the average signal value, which is normally
zero), provided the signal power is small, in a certain sense, compared to
the noise.
Consider the function space associated with these ensembles having n
dimensions. The white noise corresponds to a spherical Gaussian distribu-
tion in this space. The signal ensemble corresponds to another probability
distribution, not necessarily Gaussian or spherical. Let the second moments
of this distribution about its center of gravity be an. That is, if
p{xi y • ' • i Xn) is the density distribution function
dij = /
/ p(xi
oii)\Xj — aj) dxi ,
) a^n
where the a,- are the coordinates of the center of gravity. Now a^ is a posi- ,
tive definite quadratic form, and we can rotate our coordinate system to .
align it with the principal directions of this form, aij is then reduced to
diagonal form bu . We require that each ba be small compared to N, the
squared radius of the spherical distribution.
In this case the convolution of the noise and signal produce a Gaussian
distribution whose corresponding quadratic form is
I
The entropy power of this distribution is
[n(N + bii)]
l/n
or approximately
n— lil/n
= KNT + 2^bu(NT-']
N-\-~ Zbu .
n
The last term is the signal power, while the first is the noise power,
MATHEMATICAL THEORY OF COMMUNICATION 637
PART IV: THE CONTINUOUS CHANNEL
23. The Capacity of a Continuous Channel
In a continuous channel the input or transmitted signals will be con-
tinuous functions of tinie/(/) belonging to a certain set, and the output or
received signals will be perturbed versions of these. We will consider only
the case where both transmitted and received signals are limited to a certain
band W, They can then be specified, for a time T", by 2TW numbers, and
their statistical structure by finite dimensional distribution functions.
Thus the statistics of the transmitted signal will be determined by
I
and those of the noise by the conditional probability distribution
^xi..--. xSyi 7 "' yyn) ^ Pxiy)-
The rate of transmission of information for a continuous channel is defined
in a way analogous to that for a discrete channel, namely
R = E{x) - Hy{x)
where H{x) is the entropy of the input and Hy(x) the equivocation. The
channel capacity C is defined as the maximum of R when we vary the input
over all possible ensembles. This means that in a finite dimensional ap-
proximation we must vary P{x) = P(xi , • ■ • , Xn) and maximize
- I P{x) log P{x) dx + jj Pix, y) log ^|l|^ dx dy.
This can be written
// ^(^' y^ >°^ S^p% '^ 'y
using the fact that / / P(x^ y) log F(x) dx dy = I P(x) log P(x) dx. The
channel capacity is thus expressed
C - Lim Max |; f (p{x, y) log ^*% dx dy.
r-oo p(xy T J J ° P{x)P{y)
It is obvious in this form that R and C are independent of the coordinate
P{x, y)
system since the numerator and denominator in log T^rTTTT^ will be multi-
P(x}P{y)
plied by the same factors when x and y are transformed in any one to one
way. This integral expression for C is more general than H(x) — Hy{x).
Properly interpreted (see Appendix 7) it will always exist while H{x) — Hy(x)
638 BELL SYSTEM TECHNICAL JOURNAL
may assume an indeterminate form co — qo in some cases. This occurs, for
example, if x is limited to a surface of fewer dimensions than n in its n dimen-
sional approximation.
If the logarithmic base used in computing H{x) and Hy(x) is two then C
is the maximum number of binary digits that can be sent per second over the
channel with arbitrarily small equivocation, just as in the discrete case.
This can be seen physically by dividing the space of signals into a large num-
ber of small cells, sufficiently small so that the probability density Px(y)
of signal x being perturbed to point y is substantially constant over a cell
(either of xor y). If the cells are considered as distinct points the situation
is essentially the same as a discrete channel and the proofs used there will
apply* But it is clear physically that this quantizing of the volume into
individual points cannot in any practical situation alter the final answer
significantly, provided the regions are sufficiently small. Thus the capacity
will be the limit of the capacities for the discrete subdivisions and this is
just the continuous capacity defined above.
On the mathematical side it can be shown first (see Appendix 7) that if u
is the message, x is the signal, y is the received signal (perturbed by noise)
and V the recovered message then '
H{x) - Hy(x) > H{u) " B^{m)
regardless of what operations are performed on u to obtain :r or on y to obtain
V. Thus no matter how we encode the binary digits to obtain the signal, or
how we decode the received signal to recover the message, the discrete rate
for the binary digits does not exceed the channel capacity we have defined.
On the other hand, it is possible under very general conditions to find a
coding system for transmitting binary digits at the rate C with as small an
equivocation or frequency of errors as desired. This is true, for example, if,
when we take a finite dimensional approximating space for the signal func-
tions, P{Xj y) is continuous in both x and y except at a set of points of prob-
ability zero. *
An important special case occurs when the noise is added to the signal
and is independent of it (in the probability sense) . Then Px{y) is a function
only of the difference n = (y ~ x)j
P^{y) = Qiy - x)
and we can assign a definite entropy to the noise (independent of ^he sta-
tistics of the signa!), namely the entropy of the distribution Q(n). This
entropy will be denoted by H(n).
Theorem 16: If the signal and noise are independent and the received
signal is the sum of the transmitted signal and the noise then the rate of
MATHEMAriCAL THEORY Of COMMUNICATION 630
transmission is
R = H{y) - H(n)
i.e., the entropy of the received signal less the entropy of the noise. The
channel capacity is
C = Msix H(y)- II{n).
P(ar)
We have, since y = x -]- n:
m
H(xy y) = H{Xy «),
Expanding the left side and using the fact that x and n are independent
U{y) + Hy{:x) = n{x) + H(n)
Hence
R = H{x) - Hy(x) = H(y) - ^(w).
Since H{n) is independent of P(x), maximizing R requires maximizing
H(y) , the entropy of the received signal. If there are certain constraints on
the ensemble of transmitted signals, the entropy of the received signal must
be maximized subject to these constraints.
24. Channel Capacity with an Average Power Limitation
A simple application of Theorem 16 is the case where the noise is a white
thermal noise and the transmitted signals are limited to a certain average
power P, Then the received signals have an average power F -{- N where
N is the average noise power. The maximum entropy for the received sig-
nals occurs when they also form a white noise ensemble since this is the
greatest possible entropy for a power P -\- N and can be obtained by a
suitable choice of the ensemble of transmitted signals, namely if they form a
white noise ensemble of power P, The entropy (per second) of the re-
ceived ensemble is then
H{y) = W log 2ire(P + N),
and the noise entropy is
n{n) = W log IweN.
The channel capacity is
P -\- N
C - H(y) - H(n) = W log ^—^ .
Summarizing we have the following:
Theorem 17: The capacity of a channel of band W perturbed by white
640 BELL SYSTEM TECHNICAL JOURNAL
thermal noise of power N when the average transmitter power is P is given by
P -¥ N
This means of course that by sufficiently involved encoding systems we
P -\- N
can transmit binary digits at the rate W log2 — ^ — bits per second, with
arbitrarily small frequency of errors. It is not possible to transmit at a
higher rate by any encoding system without a definite positive frequency of
errors.
ti To Approximate this limiting rate of transmission the transmitted signals
must approximate, in statistical properties, a white noise.^ A system which
approaches the ideal rate may be described as follows: Let M = 2^ samples
of white noise be constructed each of duration T. These are assigned
binary numbers from to {M — 1). At the transmitter the message se-
quences are broken up into groups of s and for each group the corresponding
noise sample is transmitted as the signal. At the receiver the M samples are
known and the actual received signal (perturbed by noise) is compared with
each of them. The sample which has the least R.M .S . discrepancy from the
received signal is chosen as the transmitted signal and the corresponding
binary number reconstructed. This process amounts to choosing the most
probable {a posteriori) signal. The number M of noise samples used will
depend on the tolerable frequency € of errors, but for almost all selections of
samples we have
T • T • log ikr(e, T) „, , P ■\- N
Lma Lim r ^' = W log — -^- ,
so that no matter how small t is chosen, we can, by taking T sufficiently
P ^ N .
large, transmit as near as we wish to TW log — r= — binary digits in the
iV
time T. ■ "
P -{- N
Formulas similar to C = W^ log — - — - for the white noise case have
been developed independently by several other writers, although with some-
what different interpretations. We may mention the work of N. Wiener,
W' . G. Tuller, and H. Sullivan in this connection.
In the case of an arbitrary perturbing noise (not necessarily white thermal
noise) it does not appear that the maximizing problem involved in deter-
• ■
•This and other properties of the white noise case are discussed from the geometrical
point of view in "Communication in the Presence of Noise," loc. cit.
'"Cybernetics," loc. cit.
•Sc. D. thesis, Department of Electrical Engineering, M.I.T., 1948.
MATHEMATICAL THEORY OF COMMUNICATION 641
mining the channel capacity C can be solved explicitly. However, upper
and lower bounds can be set for C in terms of the average noise power N
and the noise entropy power iVi . These bounds are sufficiently close to-
gether in most practical cases to furnish a satisfactory solution to the
problem.
Theorem 18: The capacity of a channel of band W perturbed by an arbi-
trary noise is bounded by the inequalities
where
P = average transmitter power
N = average noise power
Ni = entropy power of the noise.
Here again the average power of the perturbed signals will be P + iV,
The maximum entropy for this power would occur if the received signal
were white noise and would be W log 2ire{P ~\- N), It may not be possible
to achieve this; i.e. there may not be any ensemble of transmitted signals
which, added to the perturbing noise, produce a white thermal noise at the
receiver, but at least this sets an upper bound to H{y). We have, therefore
C = max H{y) — H{n)
< Wlqg lireiP -\- N) ~ W log lireNi .
This is the upper limit given in the theorem. The lower limit can be ob-
tained by considering the rate if we make the transmitted signal a white
noise, of power P. In this case the entropy power of the received signal
must be at least as great as that of a white noise of power P -{- Ni since we
have shown in a previous theorem that the entropy power of the sum of two
ensembles is greater than or equal to the sum of the individual entropy
powers. Hence
max E(y) > W log 27re{P + TVi)
and
»
C > Wlog IweiP + Ni) - W log lireNi
As P increases, the upper and lower bounds approach each other, so we
have as an asymptotic rate
642 BELL SYSTEM TECHNICAL JOURNAL
If the noise is itself white, iV = iVi and the result reduces to the formula
proved previously:
C = W\og(l+^
If the noise is Gaussian but with a spectrum which is not necessarily flat,
Ni is the geometric mean of the noise power over the various frequencies in
the band W. Thus
f
Ni = exp 1 j^ log N{f) df
where N{f) is the noise power at frequency/.
Theorem IP: If we set the capacity for a given transmitter power P
equal to
■ P+ N - 71
C = W\og
then ij is monotonic decreasing as P increases and approaches as a limit.
Suppose that for a given power Pi the channel capacity is
TF log ^^ + ^ - "^
This means that the best signal distribution, say p{x)y when added to the
noise distribution q{x), gives a received distribution r{y) whose entropy
power is (Pi + iV — i?i). Let us increase the power to Pi + AP by adding
a white noise of power LP to the signal. The entropy of the received signal
is now at least
H{y) ^ W log 27r^(Pi + AT - t?i + AP)
by application of the theorem on the minimum entropy power of a sum.
Hence, since we can attain the // indicated, the entropy of the maximizing
distribution must be at least as great and tj must be monotonic decreasing.
To show that ?; ^ as P ^ =» consider a signal which is a white noise with
a large P. Whatever the perturbing noise, the received signal will be
approximately a white noise, if P is sufficiently large, in the sense of having
an entropy power approaching P -\- N.
25. The Channel Capacity with a Peak Power Limitation
In some applications the transmitter is limited not by the average power
output but by the peak instantaneous power. The problem of calculating
the channel capacity is then that of maximizing (by variation of the ensemble
of transmitted symbols)
H{y) - H{n)
MATHEMATICAL THEORY OF COMMUNICATIO.X
643
subject to the constraint that all the functions /(/) in the ensemble be less
than or equal to V^j say, for all /. A constraint of this type does not work
out as well mathematically as the average power limitation. The most we
5
have obtained for this case is a lower bound valid for all — , an "asymptotic"
s\ S
upper band I valid for large — I and an asymptotic value of C for — small.
Theorem 20: The channel capacity C for a band W perturbed by white
thermal noise of power N is bounded by
C ^ W log
2 S
irc^iV'
S
where S is the peak allowed transmitter power. For sufficiently large —
c < w
2
ire
S-\- N
N
(1 + e)
where e is arbitrarily small.
N
(and provided the band W starts
at 0)
C^Wlog(l +
S
N
S
We wish to maximize the entropy of the received signal. If ^ is large
this will occur very nearly when' we maximize the entropy of the trans-
mitted ensemble.
The asymptotic upper bound is obtained by relaxing the conditions on
the ensemble. Let us suppose that the power is limited to S not at every
instant of time, but only at the sample points. The maximum entropy of
the transmitted ensemble undef these weakened conditions is certainly
greater than or equal to that under the original conditions. This altered
problem can be solved easily. The maximum entropy occurs if the different
sam.ples are independent and have a distribution function which is constant
from — ^ys to + V -5. The entropy can be calculated as
W log AS.
■y
The received signal will then have an entropy less than
Wiog(AS + IweNXl +€)
644 BELL SYSTEM TECHNICAL JOURNAL
S
with € — > as TTz^ -^ 00 and the channel capacity is obtained by subtracting
N
the entropy of the white noise, W log lireN
2
- 5 + iV
W log (45 + 27reiV)(l + e) - TF log {2ireN) = PF log ^^ (1 + e).
This is the desired upper bound to the channel capacity.
To obtain a lower bound consider the same ensemble of functions. Let
these functions be passed through an ideal filter with a triangular transfer
characteristic. The gain is to be unity at frequency and decline linearly
down to gain at frequency W. We first show that the output functions
of the filter have a peak power limitation S at all times (not just the sample
sm - ., , , r
points). First we note that a pulse " going into the filter produces
2irWt
1 sin irWt
2 (irWiy
in the output. This function is never negative. The input function (in
the general case) can be thought of as the sum of a series of shifted functions
sin 2irWl
^ 2irWt
where a, the amplitude of the sample, is not greater than \/5. Hence the
output is the sum of shifted functions of the non-negative form above with
the same coefficients. These functions being non-negative, the greatest
positive value for any / is obtained when all the coefficients a have their
maximum positive values, i.e. \/S. In this case the input function was a
constant of amplitude \/S and since the filter has unit gain for D.C., the
output is the same. Hence the output ensemble has a peak power S,
The entropy of the output ensemble can be calculated from that of the
input ensemble by using the theorem dealing with such a situation. The
output entropy is equal to the input entropy plus the geometrical mean
gain of the filter;
w
'Q
log GUf =- j log ( ^„, ' ) df 2W
W
Hence the output entropy is
4S
W log 4S - 2W = W log
e^
MATHEMATICAL THEORY OF COMMUNICATION 645
and the channel capacity is greater than
2 S
W log — i irv .
. • S
We now wish to show that, for small — (peak signal power over average
white noise power), the channel capacity is approximately
C = IF log 1 + ~
(-!)■
More precisely C/W log [ 1 + ^)~^ 1 as — — > 0. Since the average signal
S
power P is less than or equal to the peak 5, it follows that for all —
C < W log (l + J) < riog (1+ I
Therefore, if we can find an ensemble of functions such that they correspond
■+i
to a rate nearly W log I 1 + tv 1 and are limited to band W and peak S the
result will be proved. Consider the ensemble of functions of the following
type. A series of / samples have the same value, either + \/5 or — \/5,
then the next / samples have the same value, etc. The value for a series
is chosen at random, probability | for -j-\rS and | for —\/S. K this
ensemble be passed through a filter with triangular gain characteristic (unit
gain at D.C.)j the output is peak limited to =b5. Furthermore the average
power is nearly S and can be made to approach this by taking / sufficiently
large. The entropy of the sum of this and the thermal noise can be found
by applying the theorem on the sum of a noise and a small signal. This
theorem will apply if
N
s
is sufficiently small. This can be insured by taking — small enough (after
iV
t is chosen). The entropy power will be 5" + iV to as close an approximation
as desired, and hence the rate of transmission as near as we wish to
W log ('±^
646 BELL SYSTEM TECHNICAL JOURNAL
PART V: THE RATE FOR A CONTINUOUS SOURCE
26. Fidelity Evaluation Functions
In the case of a discrete source of information we were able to determine a
definite rate of generating information, namely the entropy of the under-
lying stochastic process. With a continuous source the situation is con-
siderably more involved. In the first place a continuously variable quantity
can assume an infinite number of values and requires, therefore, an infinite
number of binary digits for exact specification. This means that to transmit
the output of a continuous source with exacl recovery at the receiving point
requires, in general, a channel of infinite capacity (in bits per second).
Since, ordinarily, channels have a certain amount of noise, and therefore a
finite capacity, exact transmission is impossible.
This, however, evades the real issue. Practically, we are not interested
in exact transmission when we have a continuous source, but only in trans-
mission to within a certain tolerance. The question is, can we assign a
definite rate to a continuous source when we require only a certain fidelity
of recovery, measured in a suitable way. Of course, as the fidelity require-
ments are increased the rate will increase. It will be shown that we can, in
very general cases, define such a rate, having the property that it is possible,
by properly encoding the information, to transmit it over a channel whose
capacity is equal to the rate in question, and satisfy the fidelity requirements.
A channel of smaller capacity is insufficient.
It is first necessary to give a general mathematical formulation of the idea
of fidelity of transmission. Consider the set of messages of a long duration,
say T seconds. The source is described by giving the probability density,
in the associated space, that the source will select the message in question
P{x). A given communication system is described (from the external point
of view) by giving the conditional probability Px(y) that if message x is
produced by the source the recovered message at the receiving point will
be y. The system as a whole (including source and transmission system)
is described by the probability function P{Xy y) of having message x and
final output y. If this function is known, the complete characteristics of
the system from the point of view of fidelity are known. Any evaluation
of fidelity must correspond mathematically to an operation applied to
P{Xj y). This operation must at least have the properties of a simple order-
ing of systems; i.e. it must be possible to §ay of two systems represented by
Pi{Xy y) and P2{x, y) that, according to our fidelity criterion, either (1) the
first has higher fidelity, (2) the second has higher fidelity, or (3) they have
MATHEMATICAL THEORY OF COMMUNICATION 647
equal fidelity. This means that a criterion of fidelity can be represented by
a numerically valued function:
*
v{P(x, y))
whose argument ranges over possible probability functions P{x, y).
We will now show that under very general and reasonable assumptions
the function v(F{x, y)) can be written in a seemingly much more specialized
form, namely as an average of a function p{xj y) over the set of possible values
of X and y:
v{P(x, y)) = j j P{x, y) p{x, y) dx dy
To obtain this we need only assume (1) that the source and system are
ergodic so that a very long sample will be, with probability nearly 1, typical
of the ensemble, and (2) that the evaluation is "reasonable" in the sense
that it is possible, by observing a typical input and output xi and yi , to
form a tentative evaluation on the basis of these samples; and if these
samples are increased in duration the tentative evaluation will, with proba-
bility 1, approach the exact evaluation based on a full knowledge of Pix^ y).
Let the tentative evaluation be fy(Xj y). Then the function p(x, y) ap-
proaches (as T" ^ 2c ) a constant for almost all (x, y) which are in the high
probability region corresponding to the system:
p{x, y) -^ v{P{x, y))
and we may also write
(x, y)-^ j j A'^i y)p(x, y) dx, dy
smce
// P(x, y)
dy = 1
This establishes the desired result.
The function p(x, y) has the general nature of a "distance" between x
and y. It measures how bad it is (according to our fidelity criterion) to
receive y when .r is transmitted. The general result given above can be
restated as follows: Any reasonable evaluation can be represented as an
average of a distance function over the set of messages and recovered mes-
sages X and y weighted according to the probability P{Xj y) of getting the
pair in question, provided the duration T of the messages be taken suffi-
ciently large.
'It is not a "metric" in the strict sense, however, since in general it does not satisfy
either p{x, y) - p{y, x) or pC.r, y) + piy, 2) > p(x, z).
648 BELL SYSTEM TECHNICAL JOURNAL
The following are simple examples of evaluation functions:
1. R.M.S. Criterion. •
V = {x{l) — y(i})
In this very commonly used criterion of fidelity the distance function
p(x^ y) is (apart from a constant factor) the square of the ordinary
euclidean distance between the points x and y in the associated function
space
1 r^
p(x, y) ^ 1^ I boU) - yii)\ dt
2. Frequency weighted R.M.S. criterion. More generally one can apply
different weights to the different frequency components before using an
R.M.S. measure of fidelity. This is equivalent to passing the difference
x{0 — y{l) through a shaping filter and then determining the average
power in the output. Thus let
e{i) = x(t) — y(t)
*
and
00
fit) = f e{T)k(t - r)
J— to
dt
then
p{x, y) ^^j /(O
dt
3. Absolute error criterion.
p{x, y) = - j I x{t) - y{i) | dt
4. The structure of the ear and brain determine implicitly an evaluation, or
rather a number of evaluations, appropriate in the case of speech or music
transmission. There is, for example, an "intelligibility" criterion in
which p(x, y) is equal to the relative frequency of incorrectly interpreted
words when message x{i) is received as y{l). Although we cannot give
an explicit representation of p(x, y) in these cases it could, in principle,
be determined by sufficient experimentation. Some of its properties
follow from well-known experimental results in hearing, e.g., the ear is
relatively insensitive to phase and the sensitivity to amplitude and fre-
quency is roughly logarithmic.
5. The discrete case can be considered as a specialization in which we have
MATHEMATICAL THEORY OF COMMUNICATION 649
tacitly assumed an evaluation based on the frequency of errors. The
function p{xj y) is then defined as the number of symbols in the sequence
y differing from the corresponding symbols in x divided by the total num-
ber of symbols in x.
27, The Rate for a Source Relative to a Fidelity Evaluation
We are now in a position to define a rate of generating information for a
continuous source. We are given P{x) for the source and an evaluation c
determined by a distance function p(^, y) which will be assumed continuous
in both X and y. With a particular system P(x, y) the quality is measured by
» — / / /)(.v, y) P(xy y) dx dy
Furthermore the rate of flow of binary digits corresponding to P{x^ y) is
i? = // PC, ,) log 1^ ^. <fy.
P{x)P{y)
We define the rate Ri of generating information for a given quality Vi of
reproduction to be the minimum of R when we keep v fixed at vi and vary
Px{y), That is:
R, = Min /[ P(x, y) log ^^^, dx dy
p,f«> JJ Pix}P{y)
subject to the constraint:
vi = 11 P{x, y)p(x, y) dx dy.
This means that we consider, in effect, all the communication systems that
might be used and that transmit with the required fidelity. The rate of
transmission in bits per second is calculated for each one and we choose that
having the least rate. This latter rate is the rate we assign the source for
the fidelity in question.
The justification of this definition lies in the following result:
Theorem 21: If a source has a rate Ri for a valuation Vi it is possible to
encode the output of the source and transmit it over a channel of capacity C
with fidelity as near ^i as desired provided ^i < C. This is not possible
if i?i > C.
The last statement in the theorem follows immediately from the definition
of Ri and previous results. If it were not true we could transmit more than
C bits per second over a channel of capacity C. The first part of the theorem
is proved by a method analogous to that used for Theorem 11. We may, in
the first place, divide the (:*:, y) space into a large number of small cells and
650 BELL SYSTEM TECHNICAL JOURNAL
represent the situation as a discrete case. This will not change the evalua-
tion function by more than an arbitrarily small amount (when the cells are
very small) because of the continuity assumed for p{x, y). Suppose that
Pi{Xj y) is the particular system which minimizes the rate and gives Rx . We
choose from the high probability y's a set at random containing
members where € — > as J" — > oo . With large T each chosen point will be
connected by a high probability line (as in Fig. 10) to a set of x's. A calcu-
lation similar to that used in proving Theorem 11 shows that with large T
almost all ^'s are covered by the fans from the chosen y points for almost
all choices of the y's. The communication system to be used operates as
follows: The selected points are assigned binary numbers. When a message
X is originated it will (with probability approaching 1 as Z* — > oo ) lie within
one at least of the fans. The corresponding binary number is transmitted
(or one of them chosen arbitrarily if there are several) over the channel by
suitable coding means to give a small probability of error. Since Ri ^ C
this is possible. At the receiving point the corresponding y is reconstructed
and used as the recovered message.
The evaluation vi for this system can be made arbitrarily close to vi by
taking T sufficiently large. This is due to the fact that for each long sample
of message x{t) and recovered message y{f) the evaluation approaches vi
(with probability 1).
It is interesting to note that, in this system, the noise in the recovered
message is actually produced by a kind of general quantizing at the trans-
mitter and is not produced by the noise in the channel. It is more or less
analogous to the quantizing noise in P. CM.
28. The Calculation of Rates
The definition of the rate is similar in many respects to the definition of
channel capacity. In the former
R = Max f[ P{x, y) log ^^^, dx dy
Px(y) J J P{x)P{y)
with P{x) and Vi = 11 P(x, y)p(Xj y) dx dy fixed. In the latter
C = Min fj P(x, y) log ff^. dx dy
Fix) J J • P{x)i\y)
with Px{y) fixed and possibly one or more other constraints (e.g., an average
power limitation) of the form K — fs P(x^ y) X{xy y) dx dy.
MATHEMATICAL THEORY OF COMMUmCATION 651
A partial solution of the general maximizing problem for determining the
rate of a source can be given. Using Lagrange's method we consider
P(x, y) log p^^.'p^ X + M Hx, y)p{x, y) + v{x)F{x, y)\ dx dy
The variational equation (when we take the first variation on P(xj y))
leads to
Py{x) = B{x) e^^''^^'">
where X is determined to give the required fidelity and B{x) is chosen to
satisfy
BWr'"^"'"^ dx = 1
This shows that, with best encoding, the conditional probability of a cer-
tain cause for various received y, Py{x) will decline exponentially with the
distance function p(.T, y) between the ^t; and y is question.
In the special case where the distance function p(x, y) depends only on the
(vector) difference between x and y,
p{x, y) = p{x — y)'
we have
B{x)e~'''^''~'^ dx = 1
Hence B(x) is constant, say a, and
Unfortunately these formal solutions are difficult to evaluate in particular
cases and seem to be of little value. In fact, the actual calculation of rates
has been carried out in only a few very simple cases.
If the distance function p{x, y) is the mean square discrepancy between
X and y and the message ensemble is white noise, the rate can be determined.
In that case we have
R = Min \n(x) - Hy{x)] = H(x) - Max Hy(x)
with N = (x — yY. But the Max Hy{x) occurs when y — x is a white noise,
and is equal to Wi log Ixe N where Wi is the bandwidth of the message en-
semble. Therefore
R = Wi log IttcQ — Wi log lireN
where Q is the average message power. This proves the following:
652 BELL SYSTEM TECHNICAL JOURNAL
Theorem ZZ: The rate for a white noise source of power Q and band W\
relative to an R.M.S. measure of fidelity is
where N is the allowed mean square error between original and recovered
messages.
More generally with any message source we can obtain inequalities bound-
ing the rate relative to a mean square error criterion.
Theorem Z3: The rate for any source of band Wi is bounded by
IFi log ^ < i? < TFi log I
where Q is the average power of the source, Qi its entropy power and iV the
allowed mean square error.
The lower bound follows from the fact that the max Hy(x) for a given
(x — y) = N occurs in the white noise case. The upper bound results if we
place the points (used in the proof of Theorem 21) not in the best way but
at random in a sphere of radius \/Q — N.
Acknowledgments
The writer is indebted to his colleagues at the Laboratories, particularly
to Dr. H. W. Bode, Dr. J. R. Pierce, Dr. B. McMillan, and Dr, B. M. Oliver
for many helpful suggestions and criticisms during the course of this work.
Credit should also be given to Professor N. Wiener, whose elegant solution
of the problems of filtering and prediction of stationary ensembles has con-
siderably influenced the writer's thinking in this field.
APPENDIX 5
Let Si be any measurable subset of the g ensemble, and 52 the subset of
the / ensemble which gives Si under the operation T. Then
Si = i02.
Let H be the operator which shifts all functions in a set by the time X.
Then
since T is invariant and therefore commutes with H . Hence if m[S] is the
probability measure of the set S
= m[S2] = m\Si]
MATEEUATICAL THEORY OF COMMUNICATION 653
«
where the second equality is by definition of measure in the g space the
third since the/ ensemble is stationary, and the last by definition of g meas-
ure again.
To prove that the ergodic property is preserv^ed under invariant operations,
let Si be a subset of the g ensemble which is invariant under H , and let Sz
be the set of all functions / which transform into ^i. Then
H% = H^TS2 = TH^'Si = Si
so that H Si is included in Si for all X. Now, since
m[H SH = m\Si]
this implies
H 02 — O2
for all X with mlSz] 7^ 0, 1. This contradiction shows that 5i does not exist.
APPENDIX 6
The upper bound, JVg < iV"i + A^2 , is due to the fact that the maximum
possible entropy for a power A''i + N2 occurs when we have a white noise of
this power. In this case the entropy power is Ni + N2.
To obtain the lower bound, suppose we have two distributions in ;; dimen-
sions p{xi) and q(xi) with entropy powers Ni and N2. What form should
p and q have to minimize the entropy power Nz of their convolution r(xi) :
ixi) ^ j p(yi)q{xi - ji) dji .
The entropy H3 of r is given by
H3 = — I r(xi) log r(xi) dxi .
We wish to minimize this subject to the constraints
Hi = —j p{xi) log p(xi) dXi
H2 — — I q(xi) log q{xi) dxi .
We consider then
U = -j [r{x) log rix) + \p{x) log pix) + tjtq{x) log q{x)] dx
5£/ = -/ [fl + log f(x)]drix) + X[l + log p{x)]8p(x)
+ /i[l + log q{x)8q{x)\] dx.
654
BELL SYSTEM TECHNICAL JOURNAL
If p{x) is varied at a particular argument Xi — Si, the variation in r{x) is
dr{x) = q(xi — s,)
and
dU =
- I qi^i — Si) log r(xi) dxi — X log p{sd =
and similarly when q is varied. Hence the conditions for a minimum are
/ q{^i — Si) log r{xi) = — X log p(si)
I pixi - Si) log r(xi) = —fi log q(si).
If we multiply the first by p(si) and the second by q(si) and integrate with
respect to ^ we obtain
Hs = — X Hi
Hz = — /x H2
or solving for X and 11 and replacing in the equations
Hi j q(xi — Si) log r(xi) dxi = —Hs log p(si)
H<
Xi — ' Si) log r(A;i) dx
JJ.i ^f>g /(^i)
Now suppose p{xi) and g(xi) are normal
in/2
^(a;i) =
^.*
i
qixi) =
(27r)«/2
n/2
CAU 'J^iT. ^'j ^^ Jt-j
Hij
(2t)"
TT' CaL/ ^^d£y %i J^% ^j
12
Then r(A:i) will also be normal with quadratic form Ct,. If the inverses of
these forms are «»/, b^, Cjj then
Cij — Uij "T Oij '
We wish to show that these functions satisfy the minimizing conditions if
and only if aij — Kbij and thus give the minimum H^ under the constraints.
First we have
fl 1
log r{xi) = - log — I Cij
Z Ztt
Q ^^K^ I J *4't •vj
/ q(xi - Si) log r{xi) =
n . 1
r..\ — Xj^C'K- ?■ — i'^^r />••
MATHEMATICAL THEORY OF COMMUNICATION
655
This should equal
Hz Vn 1 ,
m 1 2 ^°s 2^ ' ■ ' "
^XAijSiSj
Hi
which requires Aij = --- C,>.
Hs
Hi ...
In this case A a = —- Bij and both equations reduce to identities,
H',
APPENDIX 7
The following will indicate a more general and more rigorous approach to
the central definitions of communication theory. Consider a probability
measure space whose elements are ordered pairs (jc, y). The variables Xj y
are to be identified as the possible transmitted and received signals of some
long duration T, Let us call the set of all points whose x belongs to a subset
Si of X points the strip over Si , and similarly the set whose y belongs to ^2
the strip over 52 . We divide x and y into a collection of non-overlapping
measurable subsets Xj and Yt approximate to the rate of transmission R by
iJi = ^ E P(A-. , F,) log p^^,)PiY,)
where
F(Xi) is the probability measure of the strip over X;
P{ Yi) is the probability measure of the strip over F,-
P{Xi, Yi) is the probability measure of the intersection of the strips.
A further subdivision can never decrease i?i . For let Xi be divided into
Xi= X[-\- Xi and let
PiYi) = a
P{X[) = b
[')
= c
P(Xi) = h + c
P{Xl Yi) - d
i
P{Xu Fi) = e
1, Fi) = d -\- e
Then in the sum we have replaced (for the Xi, I'l intersection)
d ~l" c d c
{d H- e) log -jT—. — r by d\og— -\- e log — .
a{b -{- c) ah ac
It is easily shown that with the limitation we have on h, c, d, e,
d ^ e
b -\- c
]d+e
^
d ^e
¥c
^
r
656 BELL SYSTEM TECHNICAL JOURNAL
and consequently the sum is increased. Thus the various possible subdivi-
sions form a directed set, with R mono tonic increasing with refinement of
the subdivision. We may define R unambiguously as the least upper bound
for the i?i and write it
This integral, understood in the above sense, includes both the continuous
and discrete cases and of course many others which cannot be represented
in either form. It is trivial in this formulation that if x and u are in one-to-
one correspondence, the rate from utoyis equal to that from x to y. If v
is any function of y (not necessarily with an inverse) then the rate from x to
y is greater than or equal to that from jic to v since, in the calculation of the
approximations, the subdivisions of y are essentially a finer subdivision of
those for v. More generally if y and v are related not functionally but
statistically, i.e., we have a probability measure space (y, v), then i?(x, v) <
R{x, y). This means that any operation applied to the received signal, even
though it involves statistical elements, does not increase R.
Another notion which should be defined precisely in an abstract f ormu-
lation of the theory is that of dimension rate," that is the average number
of dimensions required per second to specify a member of an ensemble. In
the band limited case 2W numbers per second are sufficient. A general
definition can be framed as follows. Let/„(0 be an ensemble of functions
and let pT[/a{0,//3(01 be a metric measuring the "distance" from/a to /^
over the time T (for example the R.M.S. discrepancy over this interval.)
Let iV(€, 5, T) be the least number of elements/ which can be chosen such
that all elements of the ensemble apart from a set of measure 5 are within
the distance € of at least one of those chosen. Thus we are covering the
space to within e apart from a set of small measure 5 . We define the di-
mension rate X for the ensemble by the triple limit
\o^ N(€ 8 T)
\ = LimLimLim-" ^ ' '^
2_»0 i-,0 r-»oo
rioge
This is a generalization of the measure type definitions of dimension in
topology, and agrees with the intuitive dimension rate for simple ensembles
where the desired result is obvious.